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A HISTORY OF FORMAL LOGIC • BOCHENSKI
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A HISTORY
OF FORMAL LOGIC
BY
I. M. BOCHENSKI
TRANSLATED AND EDITED BY
IVO THOMAS
UNIVERSITY OF NOTRE DAME PRESS . 1961
A HISTORY OF FORMAL LOGIC
is a revised translation by Ivo Thomas of the German edition, Formale Logik,
by J. M. Bochehski, published and copyrighted by Verlag Karl Alber,
Freiburg/Miinchen, in 1956
© 1961 by
University of Notre Dame Press, Notre Dame, Indiana
Library of Congress Catalog Card Number 58 — 14183
Printed in the U. S.A.
PREFACE TO THE GERMAN EDITION
This history of the problems of formal logic, which we believe
to be the first comprehensive one, has grown only in small part from
the author's own researches. Its writing has been made possible
by a small group of logicians and historians of logic, those above
all of the schools of Warsaw and Munster. It is the result of their
labours that the work chiefly presents, and the author offers them
his thanks, especially to the founders Jan Lukasiewicz and Heinrich
Scholz.
A whole series of scholars has been exceptionally obliging in
giving help with the compilation. Professors E. W. Beth (Amster-
dam), Ph. Boehner O.F.M. (St. Bonaventure, N.Y.), A. Church
(Princeton), 0. Gigon (Bern), D. Ingalls (Harvard), J. Lukasiewicz
(Dublin), B. Mates (Berkeley, California), E. Moody (Columbia
University, New York), M. Morard O.P. (Fribourg), C. Regamey
(Fribourg/Lausanne) and I. Thomas O.P. (Blackfriars, Oxford) have
been kind enough to read various parts of the manuscript and
communicate to me many valuable remarks, corrections and addi-
tions. Thanks to them I have been able to remove various inexacti-
tudes and significantly improve the content. Of course they bear no
responsibility for the text in its final state.
The author is further indebted for important references and
information to Mile. M. T. d'Alverny, Reader of the Department of
Manuscripts of the Bibliotheque Nationale in Paris, Dr. J. Yajda
of the Centre National de la Recherche Scientifique in Paris, Pro-
fessors L. Minio-Paluello (Oxford), S. Hulsewe (Leiden), H. Hermes
and H. Scholz (Munster i. W.), R. Feys (Louvain) and A. Badawi
(Fuad University, Cairo). Dr. A. Menne has been kind enough to
read the proofs and make a number of suggestions.
My assistant, Dr. Thomas Raber, has proved a real collaborator
throughout. In particular, I could probably not have achieved the
translation of the texts into German without his help. He has
also been especially helpful in the compilation of the Bibliography
and the preparation of the manuscript for press.
A HISTORY OF FORMAL LOGIC
In the course of my researches I have enjoyed the help of several
European libraries. I should like to name here those in Amsterdam
(University Library), Basel (University Library), Gottingen (Nie-
dersachsische Landes- und Universitatsbibliothek), Kolmar (Stadt-
bibliothek), London (British Museum and India Office Library),
Munich (Bayerische Staatsbibliothek), Oxford (Bodleian Library)
and Paris (Bibliotheque Nationale); above all the Kern-Institut in
Leiden and the institutes for mathematical logic in Louvain and
Munster which showed me notable hospitality. Finally, last but
not least, the Cantonal and University Library of Fribourg, where
the staff has made really extraordinary efforts on my behalf.
The completion of my inquiries and the composition of this book
was made materially possible by a generous grant from the Swiss
national fund for the advancement of scientific research. This
enabled me to employ an assistant and defray the costs of several
journeys, of microfilms etc. My best thanks are due to the adminis-
trators of the fund, as to all who have helped me in the work.
Since the manuscript was completed, Fr. Ph. Boehner O.F.M.
and Dr Richard Brodfiihrer, editor of the series 'Orbis Academicus',
have died. Both are to be remembered with gratitude.
I. M. BOCHENSKI
VI
PREFACE TO THE ENGLISH EDITION
A. GENERAL
In this edition of the most considerable history of formal logic
yet published, the opportunity has of course been taken to make
some adjustments seen to be necessary in the original, with the
author's full concurrence. Only in § 36, however, has the numeration
of cited passages been altered owing to the introduction of new
matter. Those changes are as follows:
German edition English edition
36.13 36.17
36.14 36.18
36.15—17 36.21—23
Other alterations that may be noted are: the closing paragraphs of
§ 15 have been more accurately suited to the group of syllogisms
which they concern; 16.17 appears here as a principle rejected, not
accepted, by Aristotle; 27.28 has been re-interpreted and the
lengthy citation dropped; a new sub-section, on the beginnings of
combinatory logic, has been added to § 49. A few further items have
been added to the Bibliography. On grounds of economy this last
has been reproduced photographically; probably such German
remarks and abbreviations as it contains will not much inconven-
ience its users.
A word needs to be said about § 5 A, 'Technical Expressions',
which has naturally had to be largely re-written. In the orginal.
the author expressed his intention of using 'Aussage' as a name for
sentential expressions, and so for certain dispositions of black ink,
or bundles of sound-waves. So understood, the word could be
treated as generally synonymous writh the Scholastic 'proposilio'
and the English 'sentence' when used in an equally technical sense,
and was deemed a tolerable translation of Russell's 'proposition' the
reference of which is often ambiguous. But these equations evidently
cannot be maintained here; for one thing, they would warrant the
change of 'proposition' to 'sentence' throughout quotations from
vn
A HISTORY OF FORMAL LOGIC
Russell; secondly we prefer, with A. Church (1.01), to speak of
'propositional logic' rather than 'sentential logic' ; and thirdly,
one risks actual falsification of one's material by imposing on it a
grid of sharp distinctions which for the most part belongs to a
later period than anything here treated. As noted in the body of
the work, the Stoics and Frege were alone in making the distinction
between sign and significate as sharply as is now customary. So we
have normally used 'proposition' where the author used 'Aussage' -
always, indeed, when the Scholastic 'propositio' needs translation.
In Part V usage is of course largely conditioned by the fact that so
many citations now appear in their original language.
B. ABELARD
As to the contents, an evident lacuna is the absence of any texts
from the 12th century a.d., and the author himself has suggested
that 30.03 is quite insufficient reference to Peter Abelard, described
in an epitaph as 'the Aristotle of our time, the equal or superior of
all logicians there have been' (1.02), and in similar words by John
of Salisbury (1.03). We propose only to elaborate that single reference,
by way of giving a taste of this twelfth century logic, closely based
on Boethius in its past, growing in an atmosphere of keen discus-
sion in its present, evidently holding the seeds of later Scholastic
developments as exemplified in this book.
Abelard's consequences are certainly not fully emancipated from
the logic of terms (cf. 30.03) yet he realizes that propositions are
always involved. His explanation of 'consequential' may first be
noted :
1.04 A hypothetical proposition is called a 'consequence'
after its consequent, and a 'conditional' after its condition.
Speaking later of a form of the laws of transposition (43.33) he
says:
1.05 My opinion is that while the force of the inference lies
in the terms, yet the whole proposition is to be denied. . . .
Rightly the whole sequent and antecedent proposition is to
be denied, since the inference lies between the entire proposi-
tions, though the force of the inference depends on the
terms. ... So that the hypothetical proposition is rightly
said not to be composed of simple terms, but to be conjoined
from several propositions, inasmuch as it propounds that what
the sequent proposition manifests, follows from what the
precedent (manifests). So that the denial is not to be effected
according to the terms alone, but according to the entire
propositions between which the relation of consequence is
propounded.
viii
ABELARD
Consequences themselves are distinguished from their metalogical
formulations (cf. the commentary preceding 31.14), the latter being
called 'maximae propositiones' and defined thus:
1.06 That proposition which contains the sense of many
consequences and manifests the manner of proof common to
their determining features (differentiae) according to the
force of their relationship, is called a 'maximal proposition1.
E.g. along with these consequences: 'if it is man, it is animal',
'if it is pearl, it is stone', 'if it is rose, it is flower', 'if it is
redness, it is colour' etc., in which species precede genera,
a maximal proposition such as the following is adduced : of
whatever the species is predicated, the genus too (is predi-
cated). . . . This maximal proposition contains and expresses
the sense of all such consequences and manifests the way of
yielding inference common to the antecedents.
There is a rich store of maxims in Abelard, but it is not always
easy to see whether they belong to the logic of terms or propositions.
This ambiguity has been noted with reference to Kilwardby (cf.
§ 31 , B ) where one might be tempted to think that it was unconscious.
But the terminology is not subject to direct attention in Kilwardby;
in Abelard it is, and the ambiguity is noted and accepted. The
following passage may need apology for its length, but not for its
great interest in respect of terminology, semantic considerations
(on which we cannot here delay), maxims both of validity and
invalidity, and the reduction of some of them to others, 1.06 and
1.07 are enough to establish the basis and essentials of § 31 firmly
in the 12th century.
1.07 'Antecedent' and 'consequent' are sometimes used to
designate complete enunciations as when in the consequence :
if Socrates is man, Socrates is animal, we say that the first
categorical is antecedent to the second; sometimes in the
designation of simple terms (dictio) or what they signify, as
when we say in regard to the same consequence that the
species is antecedent to the genus, i.e. 'man' to 'animal', the
nature or relationship provides inferential force. . . . But
whether we take 'antecedent' and 'consequent' for simple
terms or complete enunciations, we can call them the parts of
hypothetical enunciations, i.e. the parts of which the conse-
quences are composed and of which they consist, not parts
of which they treat. For we cannot accept as true this conse-
quence: if he is man, he is animal, if it treats of utterances
(vocibus) be they terms or propositions. For it is false that if
ix
A HISTORY OF FORMAL LOGIC
the utterance 'man' exists, there should also be the utterance
'animal'; and similarly in the case of enunciations or their
concepts (intelledibus). For it is not necessary that he who has
a concept generated by the precedent proposition should also
have one generated from the consequent. For no diverse con-
cepts are so akin that one must be possessed along with the
other; indeed everyone's own experience will convince him
that his soul does not retain diverse concepts and will find
that it is totally occupied with each single concept while he
has it. But if someone were to grant that the essences of
concepts follow on one another like the essences of the
things from which the concepts are gained, he would have
to concede that every knower has an infinity of concepts
since every proposition has innumerable consequences.
Further, whether we treat of enunciations or of their concepts,
we have to use their names in a consequence; but if 'man' or
'animal' are taken as names either of enunciations or concepts,
'if there is man there is animal' cannot at all be a consequence,
being composed entirely of terms, as much as to say: 'if man
animal' ; indeed as a statement it is quite imperfect. To keep,
therefore, a genuine relation of consequence we must concede
that it is things which are being treated of, and accept the
rules of antecedent and consequent as given in the nature
of things. These rules are as follows :
(1) on the antecedent being posited, the consequent is posited;
(2) on the consequent being destroyed, the antecedent is
destroyed, thus:
'if there is man there is animal', 'if there is not animal there
is not man';
(3) neither if the antecedent is posited, is the consequent
destroyed,
(4) nor if the antecedent is destroyed need the consequent
be destroyed
(5) or posited, just as
(6) neither if the consequent is destroyed is the antecedent
posited,
(7) nor if the same (the consequent) is posited is it (the ante-
cedent) either posited
(8) or removed.
Since the last ( (6)-(8) ) are equivalent to the former ( (3)-(5) )
as also their affirmatives are mutually equivalent, the two
sets must be simultaneously true or false. The two first rules
ABELARD
are also in complete mutual agreement and can be derived
from one another, e.g. if it is conceded : if there is man there
is animal, it must also be conceded: 'if there is not animal
there is not man, and conversely.
When the first is true, the second will be proved true as
follows, by inducing an impossibility. Let us posit this as
true: if there is man there is animal, and doubt about this:
if there is not animal there is not man, i.e. whether 'animal'
negated negates 'man'. We shall confirm this in the following
way. Either 'animal' negated negates 'man' or negated it
admits 'man', so that it may* happen that when 'animal' is
denied of something man may exist in that thing. Suppose it
be conceded that when 'animal' is denied, man may persist;
yet it was formerly conceded that 'man' necessarily requires
'animal', viz. in the consequence: if there is man there is
animal. And so it is contingent that what is not animal, be
animal; for what the antecedent admits, the consequent
admits. . . . But this is impossible. . . .
Quite definitely propositional are the rules:
1.08 Whatever follows from the consequent (follows) also
from the antecedent;
Whatever implies the antecedent (implies) also the conse-
quent;
used in the derivation of categorical syllogisms.
While it is clear that the primary source of all this doctrine of
consequence is the De Differentiis Topicis of Boethius, we can also
see the germ of later developments in Abelard's realization that some
are deducible from others (1.07, 1.09), and his examinations of some
that he finds doubtful (1.10).
It is noteworthy that categorical syllogisms are presented entirely
by means of concrete instances and metalogical rules (regulae) —
which are not reckoned as maxims since the inferential (or impli-
cative) 'force' of the premisses is derived entirely from the disposi-
tion of the terms, is, in Abelard's terminology, 'complexional', a
term preserved in Kilwardby. Variables of the object-language are
nowhere used. Indeed, except in expositions of the Boethian
hypothetical syllogisms, the only place we find variables in Abelard
is a passage where he introduces a simple lettered diagram to help
the intuition in an original argument:
1.11 If a genus was always to be divided into proximate
species or proximate differences, every division of a genus
* Emending De Rijk's possint to possit
xi
A HISTORY OF FORMAL LOGIC
would be dichotomous - which was Boethius's view. . . . But
I remember having an objection to this on the score of (the
predicament of) relation. . . . This will be more easily seen if
we designate the members of the predicament by letters and
distinguish its arrangement by a figure like this.
Relation
D F G L
If now C and D were mutually related on the one hand,
B and C on the other, since B is prior to its species Z), while
D is together with (simul) its relative C, B would precede C;
so that B would precede both its species and its relative;
hence also itself.
There follow two more arguments to show that the system suppos-
ed figured stands or falls entire with any one of its parts.
There is no suggestion in Abelard that syncategorematica,
important for later theory of consequences, are a primary concern
of logic, the purpose of which he states as follows:
1.12 Logic is not a science of using or composing arguments,
but of discerning and estimating them rightly, why some are
valid, others invalid.
But he is puzzled about the signification, if any, of syncategore-
matica, and refers to various contemporary views:
1.13 Conjunctions and prepositions ought to have some
signification of their own. . . . What concepts are designated by
expressions of this kind, it is not easy to say. . . . Some think
that such expressions have sense but no reference (solos
intelledus generare, nullamque rem subiedam habere) as they
grant also to be the case with propositions. . . . There are also
some who make out that logicians have quite removed such
expressions from the class of significant ones. . . . The opinion
I favour is that of the grammarians who make contributions
to logic, that we should admit them as significant, but should
say that their significance lies in their determining certain
properties of the references (res) of the words governed by the
prepositions. . . . Conjunctions too, as indicating conjunction
of things, determine a property in their regard, e.g. when I
say: 'a man and a horse runs', by the conjunction 'and' I
unite them in runing, and at the same time indicate that by
the 'and'.
Xll
ABELARD
The emergence of a logic of propositions from one of terms is
exemplified in the rather sophisticated, and disputed, distinction
between propositional and term connectives :
1.14 It is to be remarked that while disjunctive connectives
are applied to the terms both of categorical and hypothetical
propositions, they seem to have a different sense in each. . . .
But some allow no difference . . . saying that there is the
same proposition when it is said: 'Socrates is healthy or (we/)
sick' and when it is said: 'Either Socrates is healthy or (aut)
he is sick', reckoning every disjunctive as hypothetical.
Again of temporal propositions, compounded by means of 'when',
which Abelard treats as conjunctives, he says:
1.15 It is evident in temporals that we should not estimate
the relation of consequence according to any force in the
relationship of terms, . . . but only in the mutual accompani-
ment (of the components).
And again, with the addition of truth-conditions:
1.16 In these (temporals) in which the relation of conse-
quence is to be reckoned nothing else than coincidence in
time . . . provided the members are true, people concede that
the consequence is true, and otherwise false.
Some 'rules' follow for the construction of consequences on this
basis. Among them the following deserves special notice :
1.17 Of whatever (hypotheticals) the antecedents are
concomitant, the consequents too (are concomitant), thus:
if when he is a man he is a doctor, when he is an animal he is
an artificer.
This is Leibniz's praeclarum Iheorema (cf. 43.37) in essentials,
though it seems impossible to say whether Abelard envisaged it in
its Leibnizian classical or its Russellian propositional form. He
explains it indeed by saying that 'as "man" is necessarily antecedent
to "animal", so "doctor" is to "artificer"', yet he clearly thinks of
it as compounded of propositions. The fact is that the two kinds of
logic were not yet perfectly clearly distinguished. A further indication
that the full generality of propositional logic had not yet been achie-
ved, though, as we have seen, it was already in the making, is that
while Abelard gives us as a consequence: if he is man and stone, he
is animal, he does not rise to: if he is man and stone, he is man.
There is evidently a vast deal more to be said both about the
prowess and the limitations of this logician, both on this and other
subjects, but we already exceed the limits of discussion proper to
this history. It is, however, certain that the serious beginnings of
Scholastic logic must be looked for in the 12th century.
IVO THOMAS
xm
ACKNOWLEDGEMENTS
Grateful acknowledgements are due to Bertrand Russell for
permission to quote from his The Principles of Mathematics
(London, 1903), and to the respective publishers for permission
to use passages from: The Dialogues of Plato, translated in
English by B. Jowett (Oxford University Press); The Works
of Aristotle, translated into English under the editorship of
J. A. Smith andW. D.Ross (Oxford University Press) ; Principia
Mathematica, by A. N. Whitehead and Bertrand Russell (Cam-
bridge University Press, 1925-27) ; Traciatus Logico-Philo-
sophicus, by L.Wittgenstein (Routledge and Kegan Paul, 1922) ;
Translations from the Philosophical Writings of Gottlob Frege,
by Peter Geach and Max Black (Basil Blackwell, Oxford, 1952) ;
to The Belknap Press of Harvard University Press, for quota-
tions from The Collected Papers of Charles Sanders Peirce, edited
by Charles Hartshorne and Paul Weiss (copyright 1933 by the
President and Fellows of Harvard University).
CONTENTS
Preface to the German edition v
Translator's preface to the English edition
A. General vii
B. Abelard viii
PART I
Introduction
§ 1. The concept of formal logic 2
§ 2. On the history of the history of logic 4
A. The beginnings 4
B. Prejudices 4
1. Thomas Reid 5
2. Kant 6
3. Prantl 6
4. After Prantl 8
C. Research in the 20th century 9
§ 3. The evolution of formal logic 10
A. Concerning the geography and chrorretogy of logic ... 10
B. How logic evolved 12
C. The varieties of logic 12
D. The unity of logic 14
E. The problem of progress 15
§ 4, Method and plan 18
A. History of problems, and documentation 18
B. Plan of the work 18
C. Character of the contents 19
§ 5. Terminology 20
A. Technical expressions 20
B. Concerning mathematico-logical symbolism .... 22
C. Typographical conventions 23
PART II
The Greek Variety of Logic
§ 6. Introduction to Greek logic 26
A. Logicians in chronological order 26
B. Periods '27
C. State of research 27
XV
CONTENTS
I. The precursors
§ 7. The beginnings 29
A. Texts 29
B. Significance 31
§ 8. Plato 33
A. Concept of logic 33
B. Approaches to logical formulae 34
C. Diaeresis 35
II. Aristotle
§ 9. The work of Aristotle and the problems of its literary history . . 40
A. Works 40
B. Problems 40
1. Authenticity 40
2. Character 41
3. Chronology 41
§ 10. Concept of logic. Semiotic 44
A. Name and place of logic 44
B. The subject-matter of logic 44
C. Syntax 46
D. Semantics 47
§ 11. The topics 49
A. Subject and purpose 49
B. Predicables 51
C. Categories 53
D. Sophistic 54
§ 12. Theory of opposition; principle of contradiction; principle of tertium
exclusum 57
A. Theory of opposition 57
B. Obversion 59
C. The principle of contradiction 60
D. The principle of tertium exclusum 62
§ 13. Assertoric syllogistic 63
A. Text 64
B. Interpretation 66
C. Structure of the syllogism 69
D. The figures and further syllogisms 70
§ 14. Axiomatization of the syllogistic. Further laws 72
A. Axiomatic theory of the system 72
B. Systems of syllogistic 75
C. Direct proof 76
D. Indirect proof 77
E. Dictum de omni et nullo 79
F. Beginnings of a metalogical system 80
G. The inventio medii 80
§ 15. Modal logic 81
A. Modalities 81
B. Structure of modal sentences 83
C. Negation and conversion 84
D. Syllogisms 85
§ 16. Non-analytic laws and rules 88
A. Two kinds of inference 89
B. Laws of class- and predicate-logic 91
C. Theory of identity 92
D. Syllogisms from hypotheses 93
E. Laws of the logic of relations 95
F. Propositional rules and laws 97
Summary 98
xvi
CONTENTS
§ 17. Theophrastus 99
A. Development and alteration of various doctrines . . . 99
B. Modal logic 101
C. Hypothetical syllogisms 103
III. The Megarian-Stoic School
§ 18. Historical survey 105
A. Thinkers and schools 105
B. Problems of literary history 107
C. Origin and nature . 108
§ 19. Concept of logic. Semiotics. Modalities 109
A. Logic 109
B. Lecta 110
C. Syntax Ill
D. Doctrine of categories 113
E. Truth 114
F. Modalities 114
§ 20. Proposilional functors 115
A. Negation 116
B. Implication 116
1. Philonian implication 117
2. Diodorean implication 117
3. 'Connexive' implication 118
4. 'Inclusive' implication 119
C. Disjunction 119
1. Complete disjunction 119
2. Incomplete disjunction 120
D. Conjunction 121
E. Equivalence 121
F. Other functors 121
§ 21. Arguments and schemes of inference 122
A. Conclusive, true and demonstrative arguments . . . . 122
B. Non-syllogistic arguments 124
C. Further kinds of argument 124
D. Schemes of inference 125
§ 22. Axiomatization. Compound arguments 126
A. The indemonstrables 126
B. Metatheorems 127
C. Derivation of compound arguments 128
D. Further derived arguments 130
§ 23. The liar 130
A. History 131
B. Formulation 131
C. Efforts at solution 132
IV. The Close of Antiquity
§ 24. Period of commentaries and handbooks 134
A. Characteristics and historical survey 134
B. The tree of Porphyry 135
C. Extension of logical technique 136
D. Fresh division of implication 137
E. Boethius's hypothetical syllogisms 139
F. Alterations and development of the categorical syllogistic . 140
G. The supposedly fourth figure 141
H. Pons asinorum 143
I. Anticipation of the logic of relations 144
Summary 144
xvii
CONTENTS
PART III
The Scholastic Variety of Logic
§ 25. Introduction to scholastic logic 148
A. State of research 148
B. Provisional periods 148
C. The problem of sources 150
D. Logic and the schools 150
E. Method 151
F. Characteristics 152
I. Semiotic foundations
§ 26. Subject-matter of logic 153
A. Basic notions of semiotics 153
B. Logic as a theory of second intentions 154
C. Formal logic as a theory of syncategorematic expressions . 156
D. Content of the works 159
§ 27. Supposition 162
A. Concept of supposition 163
B. Material and formal supposition 164
C. Simple supposition 168
D. Personal supposition 171
E. Interpretation in modern terms 173
§ 28. Ampliation, appellation, analogy 173
A. Ampliation 173
B. Appellation 175
C. Analogy 177
§ 29. Structure and sense of propositions 180
A. Division of propositions 180
B. Analysis of propositions 180
C. Analysis of modal propositions: dictum and modus . . . 182
D. Composite and divided senses 184
E. Reference of propositions 187
II. Propositional Logic
§ 30. Notion and division of consequences 189
A. Historical survey 189
B. Definition of consequence 190
C. Division of consequences 191
D. Meaning of implication 195
E. Disjunction 197
§ 31. Propositional consequences 198
A. Hypothetical propositions 198
B. Kilwardby 198
C. Albert of Saxony 199
D. Paul of Venice 205
E. Rules of consequences ut nunc 208
III. Logic of Terms
§ 32. Assertoric syllogistic 210
A. Early mnemonics 210
B. Barbara-Celarent 211
C. Barbara-Celaront 215
D. The fourth figure 216
1. Among the Latins 216
2. In Albalag 217
E. Combinatorial method 219
F. Inventio medii, pons asinorum 219
xviii
CONTENTS
G. The problem of the null class 221
1. St. Vincent Ferrer 221
2. Paul of Venice 223
3. John of St. Thomas 223
§ 33. Modal syllogistic 224
A. Albert the Great 224
B. Pseudo-Scotus 225
C. Ockham 227
D. Logic of propositions in future and past tenses .... 230
§ 34. Other formulae 231
A. Syllogisms with singular terms 232
B. Analysis of 'every' and 'some' 234
C. Exponible propositions 234
D. Oblique syllogisms 236
§ 35. Antinomies 237
A. Development 237
B. Formulation 239
1. The liar 239
2. Other antinomies 240
C. Solutions 241
1. The first twelve solutions 241
2. The thirteenth solution 244
3. The fourteenth solution 246
4. Preliminaries to the solution of Paul of Venice . . . 247
5. The solution of Paul of Venice 249
Summary 251
PART IV
Transitional Period
§ 36. The 'classical' logic 254
A. Humanism 254
B. Content 256
C. Psychologism 257
D. Leibniz 258
E. Comprehension and extension 258
F. The fourth figure and subaltern moods 259
G. Syllogistic diagrams 260
H. Quantification of the predicate 262
PART V
The Mathematical Variety of Logic
I. General Foundations
§37. Introduction to mathematical logic 266
A. Characteristics 266
B. Chronological sequence 267
C. Frege 268
D. Periods 269
E. State of research 270
F. Method 271
§ 38. Methods of mathematical logic 272
A. Logical calculus 272
1. Lull 272
2. Hobbes 273
3. Leibniz 274
XIX
CONTENTS
4. Lambert 276
5. Gergonne 277
6. Boole 278
7. Peirce 279
B. Theory of proof 280
1. Bolzano 280
2. Frege 282
C. Metalogic 284
§39. The concept of logic 286
A. The logistic position 287
1. Frege: semantics 287
2. Frege: logic and mathematics 289
3. Russell 290
4. Frege: number 291
B. Formalism 292
C. Intuitionism 293
II. The First Period
§ 40. The Boolean calculus 296
A. De Morgan 296
B. Boole 298
1. Symbolism and basic concepts 298
2. Applications 301
C. The logical sum 302
D. Inclusion 303
E. Peano 306
III. Propositional Logic
§ 41. Propositional logic: basic concepts and symbolism .... 307
A. Boole 307
B. McColl 309
C. Frege 310
1. Content and judgment 310
2. Implication 311
D. Peirce 313
E. Applications of his symbolism by Frege 314
F. Negation and sum in Frege 316
G. Peano's symbolism for propositional logic 317
H. Later development of symbolism for propositional logic . . 318
§ 42. Function, variable, truth-value 319
A. Logical form 320
B. Concept of function: Frege 320
C. Propositional function: Russell 322
D. Many-place functions 323
E. The variable 325
1. Frege 325
2. Russell 326
F. Truth-values 327
G. Truth-matrices 330
1. Peirce 330
2. Wittgenstein 331
H. Decision procedure of Lukasiewicz 333
§ 43. Propositional logic as a system 335
A. McColl 335
B. Frege's rules of inference 337
C. Propositional laws from the Begriffsschrift 338
D. Whitehead and Russell 340
1. Primitive symbols and definition 340
CONTENTS
2. Axioms (primitive propositions) 340
3. Statement of proofs 341
4. Laws 342
E. Sheffer's functor 344
F. Lukasiewicz's statement of proofs 345
IV. Logic of Terms
§ 44. Predicate logic 347
A. Quantifiers 348
1. Mitchell 348
2. Peirce . 348
3. Peano 349
4. Frege 350
B. Apparent variables 353
1. Peano 353
2. Whitehead and Russell 353
C. Formal implication 354
D. Laws of one-place predicates 355
E. Laws of many-place predicates 356
F. Identity 357
§ 45. The logic of classes 359
A. Individual and class. Concept of element 360
B. Meaning and extension 360
C. The plural article 362
D. Definition of classes by functions 363
E. Product and inclusion of classes 364
§ 46. Existence 365
A. The null class 365
B. Null class and assertoric syllogistic 366
C. Description 367
1. The definite article: Frege 367
2. Logical existence 369
3. Description in Russell 371
4. Symbolism 373
(a) Peano 373
(b) Principia 374
V. Other Doctrines
§ 47. Logic of relations 375
A. Laying the foundations 375
1. De Morgan 375
2. Peirce 377
3. Russell 379
4. Principia 380
B. Series 3S4
1. Frege 385
2. Principia 385
C. Isomorphy 386
§ 48. Antinomies and theories of types 387
A. Historical survey 3S7
B. The antinomies 388
C. Anticipations of the theory of types 391
D. The ramified theory of types 393
E. Systematic ambiguity 396
F. The axiom of reducibility 398
G. Simple theory of types 399
1. Chwistek 399
2. Ramsey 400
XXI
CONTENTS
§ 49. Some recent doctrines 402
A. Strict implication: Lewis 403
B. Many-valued logics: Lukasiewicz 405
C. Godel's theorem 407
D. Combinatory logic 411
Summary 412
PART VI
The Indian Variety of Logic
§ 50. Introduction to Indian logic 416
A. Historical survey 416
B. Evolution of formal logic 417
C. State of research 419
D. Method 420
§ 51. The precursors 421
A. Milinda-Panha 421
B. Kathavatthu 421
C. The ten-membered formula 423
§ 52. Vaisesika- and Nyaya-sutra 425
A. Vaisesika-sutra 425
1. Doctrine of categories 425
2. Inference 426
B. Nyaya-sutra 426
1. Text 426
2. Vatsyayana's commentary 428
3. Interpretation 429
§ 53. The rise of formal logic . 431
A. Main stages of development 431
B. Terminology 432
C. The three-membered syllogism 433
D. The three-membered rule: trairupya 435
E. Wheel of reasons: hetu-cakra 435
F. 'Eva' 437
G. Universal connection 437
H. Final form of the doctrine 439
§ 54. Some other logical doctrines 440
A. Apoha 441
B. Definitions of vyapti 441
C. Some basic concepts 442
D. The law of double negation 444
E. Relation logic, definition of number 445
Summary 446
REFERENCES, BIBLIOGRAPHY, INDICES
I. References 451
II. Bibliography 460
III. Indices 535
1. Index of names 535
2. Index of logical symbols 541
3. Index of mnemonics 544
4. Subject index 545
Plate I opposite p. 220
Plate II opposite p. 260
Plate III opposite p. 274
Plate IV opposite p. 316
XX11
PART I
Introduction
1 Bochenski, Formal Logic
§1. THE CONCEPT OF FORMAL LOGIC
Preliminary definition of the subject matter of the history of
logic is hard to come by. For apart from 'philosophy' there is perhaps
no name of a branch of knowledge that has been given so many
meanings as 'logic'. Sometimes the whole of philosophy, and even
knowledge in general, has been thus named, from metaphysics on
the one hand, cf. Hegel, to aesthetics ('logic of beauty') on the other,
with psychology, epistemology, mathematics etc. in between. With
such a wide choice it is quite impossible to include in a history of
logical problems all that has been termed 'logic' in the course of
western thought. To do so would practically involve writing a
general history of philosophy.
But it does not follow that the use of the name 'logic' must be
quite arbitrary, for history provides several clues to guide a choice
between its many meanings. This choice can be arrived at by the
following stages.
1. First let us discard whatever most authors either expressly
ascribe to some other discipline, or call 'logic' with the addition of
an adjective, as for example epistemology, transcendental logic,
ontology etc.
2. When we examine what remains, we find that there is one
thinker who so distinctly marked out the basic problems of this
residual domain that all later western inquirers trace their descent
from him: Aristotle. Admittedly, in the course of centuries very
many of these inquirers - among them even his principal pupil and
successor Theophrastus - have altered Aristotelian positions and
replaced them with others. But the essential problematic of their work
was, so far as we know, in constant dependence in one way or another
on that of Aristotle's Organon. Consequently we shall denote as
'logic' primarily those problems which have developed from that
problematic.
3. When we come to the post-Aristotelian history of logic, we can
easily see that one part of the Organon has exercised the most
decisive influence, namely the Prior Analytics. At some periods
other parts too, such as the Topics or the Posterior Analytics, have
indeed been keenly investigated and developed. But it is generally
true of all periods marked by an active interest in the Organon that
the problems mainly discussed are of the kind already to hand in
the Prior Analytics. So the third step brings us to the point of
describing as 'logic' in the stricter sense that kind of problematic
presented in the Prior Analytics.
4. The Prior Analytics treats of the so-called syllogism, this being
defined as aXoyo<;in which if something is posited, something else
necessarily follows. Moreover such Xoyot, are there treated as
formulas which exhibit variables in place of words with constant
CONCEPT OF FORMAL LOGIC
meaning; an example is 'B belongs to all A'. The problem evidently,
though not explicitly, presented by Aristotle in this epoch-making
work, could be formulated as follows. What formulas of the prescri-
bed type, when their variables are replaced by constants, yield
conditional statements such that when the antecedent is accepted,
the consequent must be admitted? Such formulas are called 'logical
sentences'. We shall accordingly treat sentences of this kind as a
principal subject of logic.
5. Some logicians have limited themselves to the discovery,
examination and systematic ordering of logical theorems, e.g. many
scholastic and mathematical logicians, as also Aristotle himself in
the Prior Analytics. But logic so understood seems too narrowly
conceived. For two kinds of problem naturally arise out of the
theorems. First those about their nature - are they linguistic
expressions, word-structures, psychical forms or functions, objective
complexes? What does a logical law mean, what does a statement
mean? These are problems which nowadays are dealt with in semio-
tics. Second, problems relevant to the question how logical laws
can be correctly applied to practical scientific thought. These were
dealt with by Aristotle himself, principally in the Posterior Analytics,
and nowadays are the concern of general methodology. So semiotic
and methodological problems are closely connected with logic;
in practice they are always based on semiotics and completed in
methodology. What remains over and above these two disciplines
we shall call formal logic.
6. A complete history of the problems of logic must then have
formal logic at its centre, but treat also of the development of
problems of semiotics and methodology. Before all else it must put
the question : what problems were in the past posited with reference
to the formulation, assessment and systematization of the laws of
formal logic? Beyond that it must look for the sense in which these
problems were understood by the various logicians of the past, and
also attempt to answer the question of the application of these
laws in scientific practice. We have now delimited our subject, and
done so, as we think, in accordance with historical evidence.
But such a program has proved to be beyond accomplishment.
Not only is our present knowledge of seim'otic and methodological
questions in the most important periods too fragmentary, but even
where the material is sufficiently available, a thorough treatment
would lead too far afield. Accordingly we have resolved to limit
ourselves in the main to matters of purely formal logic, giving only
incidental consideration to points" from the other domains.
Thus the subject of this work is constituted by those problems
which are relevant to the structure, interconnection and truth of
sentences of formal logic (similar to the Aristotelian syllogism).
Does it or does it not follow? And, why? How can one prove the
A HISTORY OF FORMAL LOGIC
validity of this or that sentence of formal logic? How define one
or another logical constant, e.g. 'or', 'and', 'if--then', 'every' etc.
Those are the questions of which the history will here be considered.
§2. ON THE HISTORY OF THE HISTORY
OF LOGIC
A. THE BEGINNINGS
The first efforts to write a history of logic are to be found among
the humanists, and perhaps Petrus Ramus may here be counted as
the first historian. In his Scholarum dialeclicarum libri XX we find
some thirty long colums allotted to this history. To be sure, Ramus's
imagination far outruns his logic: he speaks of a logica Patrum
in which Noah and Prometheus figure as the first logicians,
then of a logica mathematicorum which alludes to the Pythagoreans.
There follow a logica physicorum (Zeno of Elea, Hippocrates, Demo-
critus etc.), the logica Socratis, Pyrrhonis el Epicrelici (sicl), the
logica Anlislheniorum el Stoicorum (here the Megarians too are
named, among others Diodorus Cronus) and the logica Academiorum.
Only then comes the logica Peripaleticorum where Ramus mentions
what he calls the Arislotelis bibliotheca i.e. the Organon (which
according to him, as in our own time according to P. Zurcher. S.J., is
not by Aristotle), and finally the logica Aristoteleorum inter prelum el
praecipue Galeni (2.01).
This book was written in the middle of the 16th century. Some
fifty years later we find a less comprehensive but more scientific
attempt by B. Keckermann. His work (2.02) is still valuable, parti-
cularly for a large collection of accurately dated titles. It remains an
important foundation for the study of lGth century logic. But its
judgments are not much more reliable than those of Ramus. Kecker-
mann seems to have given only a cursory reading to most of the
logicians he cites, Hospinianus (2.03) for example. The book is
indeed more of a bibliography than a history of logic.
B. PREJUDICES
For all his faults, Ramus was a logician; Keckermann too had
some knowledge of the subject. The same can seldom be said of their
successors until Bolzano, Peirce and Peano. Most historians of logic
in the 17th, 18th and 19th centuries treat of ontological, epistemolo-
gical and psychological problems rather than of logical ones. Further-
more, everything in this period, with few exceptions, is so condi-
tioned by the then prevailing prejudices that we may count the
whole period as part of the pre-history of our science.
HISTORY OF THE HISTORY OF LOGIC
These prejudices are essentially three:
1. First, everyone was convinced that formalism has very little
to do with genuine logic. Hence investigations of formal logic either
passed unnoticed or were contemptuously treated as quite subsidiary.
2. Second, and in part because of the prejudice already mentioned,
the scholastic period was treated as a media tempestas, a 'dark
middle age' altogether lacking in science. But as the Scholastics
were in possession of a highly developed formal logic, people sought
in history either for quite different 'logics' from that of Aristotle
(not only those of Noe and Epictetus, as Ramus had done, but even,
after his time, that of Ramus himself), or at least for a supposedly
better interpretation of him, which put the whole investigation on the
wrong track.
3. Finally, of equal influence was a strange belief in the linear
development of every science, logic included. Hence there was a
permanent inclination to rank inferior 'modern' books higher than
works of genius from older classical writers.
1. Thomas Reid
As an example of how history was written then, we shall cite one
man who had the good will at least to read Aristotle, and who
succeeded in doing so for most parts of the Organon, though he
failed for just the most important treatises. Here are his own words
on the subject:
2.04 In attempting to give some account of the Analytics
and of the Topics of Aristotle, ingenuity requires me to con-
fess, that though I have often purposed to read the whole
with care, and to understand what is intelligible, yet my
courage and patience always failed before I had done. Why
should I throw away so much time and painful attention
upon a thing of so little real use ? If I had lived in those ages
when the knowledge of Aristotle's Organon entitled a man to
the highest rank in philosophy, ambition might have induced
me to employ upon it some years' painful study; and less, I
conceive, would not be sufficient. Such reflections as these,
always got the better of my resolution, when the first ardour
began to cool. All I can say is, that I have read some parts of
the different books with care, some slightly, and some perhaps
not at all. ... Of all reading it is the most dry and the most
painful, employing an infinite labour of demonstration, about
things of the most abstract nature, delivered in a laconic
style, and often, I think, with affected obscurity; and all to
prove general propositions, which when applied to particular
instances appear self-evident.
A HISTORY OF FORMAL LOGIC
In the first place this is a really touching avowal that Reid
lectured on the teaching of a logician whom he had not once read
closely, and, what is much more important, that for this Scottish
philosopher formal logic was useless, incomprehensible and tedious.
But beyond that, the texts that seemed to him most unintelligible
and useless are just those that every logician counts among the most
exquisite and historically fruitful.
Nearly all philosophers of the so-called modern period, from the
humanists to the rise of mathematical logic, held similar views. In
such circumstances there could be no scientific history of logic, for
that presupposes some understanding of the science of logic.
The attitude towards formal logic just described will be further
illustrated in the chapter on 'classical' logic. Here we shall delay
only on Kant, who expressed opinions directly relevant to the
history of logic.
2. Kanl
Kant did not fall a victim to the first and third of the prejudices
just mentioned. He had the insight to state that the logic of his
time — he knew no other — was no better than that of Aristotle, and
went on to draw the conclusion that logic had made no progress
since him.
2.05 That Logic has advanced in this sure course, even from
the earliest times, is apparent from the fact that since Ari-
stotle, it has been unable to advance a step, and thus to all
appearance has reached its completion. For if some of the
moderns have thought to enlarge its domain by introducing
psychological discussions . . . metaphysical ... or anthropological
discussions . . . this attempt, on the part of these authors,
only shows their ignorance of the peculiar nature of logical
science. We do not enlarge, but disfigure the sciences when we
lose sight of their respective limits, and allow them to run into
one another. Now logic is enclosed within limits which admit
of perfectly clear definition; it is a science which has for its
object nothing but the exposition and proof of the formal laws
of all thought, whether it be a priori or empirical. . . .
3. Prantl
It is a remarkable fact, unique perhaps in the writing of history,
that Carl Prantl, the first to write a comprehensive history of
western logic (2.06), on which task he spent a lifetime, did it pre-
cisely to prove that Kant was right, i.e. that formal logic has no
history at all.
His great work contains a collection of texts, often arranged from
6
HISTORY OF THE HISTORY OF LOGIC
a wrong standpoint, and no longer sufficient but still indispensable.
He is the first to take and discuss seriously all the ancient and scho-
lastic logicians to whom he had access, though mostly in a polemi-
cal and mistaken spirit. Hence one can say that he founded the
history of logic and bequeathed to us a work of the highest utility.
Yet at the same time nearly all his comments on these logicians
are so conditioned by the prejudices we have enumerated, are written
too with such ignorance of the problems of logic, that he cannot be
credited with any scientific value. Prantl starts from Kant's asser-
tion, believing as he does that whatever came after Aristotle was
only a corruption of Aristotle's thought. To be formal in logic, is in
his view to be unscientific. Further, his interpretations, even of
Aristotle, instead of being based on the texts, rely only on the
standpoint of the decadent 'modern' logic. Accordingly, for example,
Aristotelian syllogisms are misinterpreted in the sense of Ockham,
every formula of propositional logic is explained in the logic of
terms, investigation of objects other than syllogistic characterized as
'rank luxuriance', and so of course not one genuine problem of
formal logic is mentioned.
While this attitude by itself makes the work wholly unscientific
and, except as a collection of texts, worthless, these characteristics
are aggravated by a real hatred of all that Prantl, owing to his
logical bias, considers incorrect. And this hatred is extended from
the teachings to the teachers. Conspicuous among its victims are the
thinkers of the Megarian, Stoic and Scholastic traditions. Ridicule,
and even common abuse, is heaped on them by reason of just those
passages where they develop manifestly important and fruitful
doctrines of formal logic.
We shall illustrate this with some passages from his Geschichte der
Logik, few in number when compared with the many available.
Ghrysippus, one of the greatest Stoic logicians, 'really accom-
plished nothing new in logic, since he only repeats what the Peri-
patetics had already made available and the peculiarities introduced
by the Megarians. His importance consists in his sinking to handle the
material with a deplorable degree of platitude, triviality and scho-
lastic niggling', Chrysippus 'is a prototype of all pedantic narrow-
mindedness' (2.07). Stoic logic is in general a 'corruption' of that
previously attained (2.08), a 'boundless stupidity', since 'Even he
who merely copies other people's work, thereby runs the risk of
bringing to view only his own blunders' (2.09). The Stoic laws are
'proofs of poverty of intellect' (2.10). And the Stoics were not only
stupid; they were also morally bad men, because they were subtle:
their attitude has 'not only no logical worth, but in the realm of
ethics manifests a moment deficient in morality' (2.11). — Of
Scholasticism Prantl says : 'A feeling of pity steals over us when we
see how even such partialities as are possible within an extremely
A HISTORY OF FORMAL LOGIC
limited field of view are exploited with plodding industry even to the
point of exhaustion, or when centuries are wasted in their fruitless
efforts to systematize nonsense' (2.12). Consequently 'so far as
concerns the progress of every science that can properly be termed
philosophy, the Middle Ages must be considered as a lost millennium'
(2.13).
In the 13th century and later, things are no better. 'Between the
countless authors who without a single exception subsist only on the
goods of others, there is but one distinction to be made.. There are
the imbeciles such as e.g. Albertus Magnus and Thomas Aquinas,
who hastily collect ill-assorted portions of other people's wealth in a
thoughtless passion for authority; and those others, such as e.g.
Duns Scotus, Occam and Marsilius, who at least understand with
more discernment how to exploit the material at hand' (2.14).
'Albertus Magnus too . . . was a muddle-head' (2.15). To take
Thomas Aquinas 'for a thinker in his own right' would be 'a great
mistake' (2.16). His pretended philosophy is only 'his unintelligent
confusion of two essentially different standpoints; since only a
muddled mind can . . .' etc. (2.17).
Similar judgment is passed on later scholastic logic; a chapter on
the subject is headed 'Rankest Luxuriance' (2.18). Prantl regrets
having to recount the views of these logicians, 'since the only alter-
native interpretation of the facts, which would consist simply in
saying that this whole logic is a mindless urge, would be blameworthy
in a historian, and without sufficient proof would not gain credence'
(2.19).
To refute Prantl in detail would be a huge and hardly profitable
task. It is better to disregard him entirely. He must, unhappily, be
treated as non-existent by a modern historian of logic. Refutation is
in any case effected by the total results of subsequent research as
recapitulated in this book.
4. After Prantl
Prantl exercised a decisive influence on the writing of history
of logic in the 19th and to some extent also in the 20th century. Till
the rise of the new investigations deriving from circles acquainted
with mathematical logic, Prantl's interpretations and evaluations
were uncritically accepted almost entire. For the most part, too, the
later historians of logic carried still further than Prantl the mingling
of non-logical with logical questions. This can be seen in the practice
of giving a great deal of space in their histories to thinkers who were
not logicians, and leaving logicians more and more out of account.
Some examples follow. F. Ueberweg, himself no mean logician,
(he could, e.g. distinguish propositional from term-logic, a rare gift
in the 19th century), devoted four pages of his survey of the history
of logic (2.20) to Aristotle, two to 'the Epicureans, Stoics and
8
HISTORY OF THE HISTORY OF LOGIC
Sceptics', two to the whole of Scholasticism - but fifty-five to the
utterly barren period stretching from Descartes to his own day.
Therein Schleiermacher, for instance, gets more space than the
Stoics, and Descartes as much as all Scholasticism. R. Adamson
(2.21) allots no less that sixteen pages to Kant, but only five to
the whole period between the death of Aristotle and Bacon, com-
prising the Megarians, Stoics, Commentators and Scholastics. A few
years ago Max Polenz gave barely a dozen pages to Stoic logic in his
big book on this school (22.2).
Along with this basic attitude went a misunderstanding of ancient
logical teaching. It was consistently treated as though exhibiting
nothing what corresponded with the content of 'classical' logic; all
else either went quite unnoticed or was interpreted in the sense of
the 'classical' syllogistic, or again, written off as mere subtlety. It is
impossible to discuss the details of these misinterpretations, but at
least they should be illustrated by some examples.
The Aristotelian assertoric syllogistic is distortedly present in
'classical' i.e. Ockhamist style (34.01) as a rule of inference with the
immortal 'Socrates' brought into the minor premiss, whereas for
Aristotle the syllogism is a conditional propositional form (§13)
without any singular terms. Stoic logic was throughout absurdly
treated as a term logic (2.23), whereas it was quite plainly a pro-
positional logic (§20). Aristotelian modal logic (§15) was so little
understood that when A. Becker gave the correct interpretation of its
teaching in 1934 (2.24), his view was generally thought to be revo-
lutionary, though in essence this interpretation is quite elementary
and was known to Albert the Great (33.03). Aristotle and Thomas
Aquinas were both credited with the Theophrastan analysis of
modal propositions and modal syllogisms, which they never advo-
cated (2.25).
No wonder then that with the rise of mathematical logic theorems
belonging to the elementary wealth of past epochs were saddled with
the names of De Morgan, Peirce and others; there was as yet no
scientific history of formal logic.
C. RESEARCH IN THE 20th CENTURY
Scientific history of formal logic, free from the prejudices we have
mentioned and based on a thorough study of texts, first developed in
the 20th century. The most important researches in the various
fields are referred to in the relevant parts of our survey. Here we
shall only notice the following points.
The rise of modern history of logic concerning all periods save
the mathematical was made possible by the work of historians
of philosophy and philologists in the 19th century. These published
for the first time a series of correct texts edited with reference to
their context in the history of literature. But the majority of ancient
9
A HISTORY OF FORMAL LOGIC
philologists, medievalists and Sanskrit scholars had only slight
understanding of and little interest in formal logic. History of
logic could not be established on the sole basis of their great and
laborious work.
For its appearance we have to thank the fact that formal logic
took on a new lease of life and was re-born as mathematical. Nearly
all the more recent researches in this history were carried out by
mathematical logicians or by historians trained in mathematical
logic. We mention only three here: Charles Sanders Peirce, the fore-
runner of modern research, versed in ancient and scholastic logic;
Heinrich Scholz and Jan Lukasiewicz, with their publications of
1931 and 1935 (2.26, 2.27), both exercising a decisive influence
on many parts of the history of logic, thanks to whom there have
appeared serious studies of ancient, medieval and Indian logic.
But still we have only made a start. Though we are already in
possession of basic insights into the nature of the different historical
varieties of formal logic, our knowledge is still mostly fragmentary.
This is markedly the case for Scholastic and Indian logic. But as
the history of logic is now being systematically attended to by a
small group of researchers it can be foreseen that this state of
affairs will be improved in the coming decades.
§3.THE EVOLUTION OF FORMAL LOGIC
As an introduction to the present state of research and to justify
the arrangement of this book, a summary presentation of results is
now needed. The view we present is a new one of the growth of
formal logic, stated here for the first time. It is a view which markedly
diverges not only from all previous conceptions of the history of
logic, but also from opinions that are still widespread about the
general history of thought. But it is no 'synthetic a priori judgment',
rather is it a position adopted in accordance with empirical findings
and based on the total results of the present book. Its significance
seems not to be confined within the boundaries of the history of
logic: the view might be taken as a contribution to the general
history of human thought and hence to the sociology of knowledge.
A. CONCERNING THE GEOGRAPHY AND CHRONOLOGY OF LOGIC
Formal logic, so far as we know, originated in two and only two
cultural regions: in the west and in India. Elsewhere, e.g. in China,
we do occasionally find a method of discussion and a sophistic
(3.01), but no formal logic in the sense of Aristotle or Dignaga was
developed there.
Both these logics later spread far beyond the frontiers of their
native region. We are not now speaking merely of the extension of
10
EVOLUTION OF FORMAL LOGIC
European logic to America, Australia and other countries settled
from Europe; North America, for instance, which from the time
of Peirce has been one of the most important centres of logical
research, can be treated as belonging to the western cultural region.
Rather it is a matter of western logic having conquered the Arabian
world in the high Middle Age, and penetrated Armenian culture
through missionaries.* Other examples could be adduced. The
same holds for Indian logic, which penetrated to Tibet, China,
Japan and elsewhere. Geographically, then, we arc concerned with
two vital centres of evolution for logic, whose influence eventually
spread far abroad.
On the subject of the chronology of logic and its division into
periods there is this to be said : this history begins in Europe in the
4th century B.C., in India about the 1st century a.d. Previously
there is in Greece, India and China, perhaps also in other places,
something like a pre-history of logic; but it is a complete mistake
to speak of a 'logic of the Upanishads' or a 'logic of the Pythagoreans'.
Thinkers of these schools did indeed establish chains of inference,
but logic consists in studying inference, not in inferring. No such
study can be detected with certainty before Plato and the Nyaga;
at best we find some customary, fixed and canonical rules of dis-
cussion, but any complete critical appreciation and analysis of
these rules are missing.
The history of western logic can be divided into five periods:
1. the ancient period (to the 6th century a.d.); 2. the high Middle
Age (7th to 11th centuries); 3. the Scholastic period (11th to loth
centuries); 4. the older period of modern 'classical' logic (16th to
19th centuries); 5. mathematical logic (from the middle of the 19th
century). Two of those are not creative periods — the high Middle
Age and the time of 'classical' logic, so that they can be left almost
unnoticed in a history of problems. The hypothesis that there was
no creative logical investigation between the ancient and Scholastic
periods might very probably be destroyed by a knowledge of Arabian
logic, but so far little work has been done on this, and as the results
of what research has been undertaken are only to be found in Arabic,
they are unfortunately not available to us.
Indian logic cannot so far be divided into periods with comparable
exactness. It only seems safe to say that we must accept at least
two great periods, the older Nyaya and Buddhism up to the 10th
century of our era, and the Navya (new) Nyaya from the 12th
century onwards.
* I am grateful to Prof. M. van den Oudenrijn for having drawn my attention
to this fact.
11
A HISTORY OF FORMAL LOGIC
B. HOW LOGIC EVOLVED
Logic shows no linear continuity of evolution. Its history resembles
rather a broken line. From modest beginnings it usually raises
itself to a notable height very quickly — within about a century —
but then the decline follows as fast. Former gains are forgotten, the
problems are no longer found interesting, or the very possibility
of carrying on the study is destroyed by political and cultural
events. Then, after centuries, the search begins anew. Nothing
of the old wealth remains but a few fragments; building on those,
logic rises again.
We might therefore suppose that the evolution of logic could be
presented as a sine-curve; a long decline following on short periods
of elevation. But such a picture would not be exact. The 'new' logic
which follows on a period of barbarism is not a simple expansion of
the old ; it has for the most part different presuppositions and points
of view, uses a different technique and evolves aspects of the
problematic that previously received little notice. It takes on a
different shape from the logic of the past.
That holds in the temporal dimension for western and, with some
limitations, for Indian logic. Perhaps it also holds in the spatial
dimension for the relation between the two considered as wholes.
We can indeed aptly compare Indian logic with ancient and Scholas-
tic logic in Europe, as lacking the notion of calculation; but beyond
that there is hardly any resemblance. They are different varieties
of logic. It is difficult to fit the Indian achievements into a scheme
of evolution in the west.
The essential feature of the whole history of logic seems then to be
the appearance of different varieties of this science separated both
in time and space.
C. THE VARIETIES OF LOGIC
There are in essence, so far as we can determine, four such forms:
1. The Ancient Variety of Logic. In this period logical theorems
are mostly formulated in the object-language, and semantics is in
being, though undeveloped. The logical formulae consist of words
of ordinary language with addition of variables. But this ordinary
language is as it were simplified, in that the chief words in it occur
only in their immediate semantic function. The basis of this logic
is the thought as expressed in natural language, and the syntactical
laws of the language are presupposed. It is from this material that
the ancient logicians abstract their formal laws and rules.
2. The Scholastic Variety of Logic. The Scholastics began by linking
themselves to antiquity, and thus far simply took over and developed
what was old. But from the end of the 12th century they started to
construct something entirely new. This logic which is properly
12
EVOLUTION OF FORMAL LOGIC
their own is almost all formulated metalogically. It is based on and
accompanied by an accurate and well-developed semantics. For-
mulae consist of words from ordinary language, with very few or no
variables, but there results no narrowing of the semantic functions
as in antiquity. Scholastic logic is accordingly a thorough-going
attempt to grasp formal laws expressed in natural language (Latin
with plentifully differentiated syntactical rules and semantic
functions. As in ancient logic, so here too we have to do with
abstraction from ordinary language.
3. The Mathematical Variety of Logic. Here we find a certain
regress to the ancient variety. Till a fairly late date (about 1930;
mathematical logic is formulated purely in the object-language,
with rich use of variables; the words and signs used have narrowly
limited semantic functions; semantics remains almost unnoticed
and plays not nearly so marked a role as in the Middle Ages and
after its resurgence since about 1930. Mathematical logic introduces
two novelties; first, the use of an artifical language; second, and
more important, the constructive development of logic. This last
means that the system is first developed formalistically and only
afterwards interpreted, at least in principle.
Common to the three western varieties of logic is a far-reaching
formalism and preponderantly extensional treatment of logical laws.
4. The Indian Variety of Logic. This differs from the western in
both the characteristics just mentioned. Indian logic succeeds is
stating certain formal laws, but formalism is little developed and is
obviously considered to be subsidiary. Again the standpoint is
preponderantly intensional in so far as the Indian logicians of the
last period knew how to formulate a highly developed logic of
terms without employing quantifiers.
The fore-going arrangement is schematic and oversimplified,
especially in regard to ancient and Indian logic. One could ask, for
instance, whether Megarian-Stoic logic really belongs to the same
variety as Aristotelian, or whether it is on the contrary mainly
new, having regard to its markedly semantic attitude.
Still more justified would perhaps be the division of Indian logic
into different forms. One could find, for example, considerable
justification for saying that Buddhist logic differs notably from the
strict Nyaya tradition not only in its philosophical basis, nor only
in details, but with this big difference that the Buddhists show a
manifestly extensional tendency in contrast to the Nyaya com-
mentators. Again, evidence is not lacking that the Navy a Nyaya
does not properly exhibit a quite new type of logic, since in some
doctrines, as in the matter of Vyapti, it takes over Buddhist modes
of expression, in others it follows the Nyaya tradition, in others
again it develops a new set of problems and takes up a fresh stand-
point.
13
A HISTORY OF FORMAL LOGIC
However the difference between Aristotle and the Megarian-
Stoic school seems hardly significant enough to justify speaking of
two different forms of logic. As to Indian logic our knowledge is so
incomplete that it would be rash to draft a division and charac-
terization of its different forms.
A further problem that belongs here is that of the so-called
'classical' logic. One could understand it as a distinct variety, since
while it consists of fragments of scholastic logic (taking over for
example the mnemonic Barbara, Celarenl etc., yet these fragments
are interpreted quite unscholastically, in an ancient rather than
scholastic way. But the content of this logic is so poor, it is loaded
with so many utter misunderstandings, and its creative power is so
extremely weak, that one can hardly risk calling something so
decadent a distinct variety of logic and so setting it on a level with
ancient, scholastic, mathematical and Indian logic.
D. THE UNITY OF LOGIC
We said above that every new variety of logic contains new
logical problems. It is easy to find examples of that: in Scholasticism
there are the magnificent semiotic investigations about the proprie-
taies terminorum, then the analysis of propositions containing time-
variables, investigations about quantifiers, etc; in mathematical
logic the problems of multiple quantification, description, logical
paradoxes, and so on. It is evident that quite different systems of
formal logic are developed as a result. To be sure, that also sometimes
happens within the framework of a single form of logic, as when we
single out Theophrastan modal logic as different from Aristotle's.
The class of alternative systems of formal logic has increased
greatly especially since Principia Mathematica.
One might therefore get the impression that the history of logic
evidences a relativism in logical doctrine, i.e. that we see the rise
of different logics. But we have spoken not of different logics, rather
of different varieties of one logic. This way of speaking has been
chosen for speculative reasons, viz. that the existence of many
systems of logic provides no proof that logic is relative. There is,
further, an empirical basis for speaking of one logic. For history
shows us not only the emergence of new problems and laws but
also, and perhaps much more strikingly, the persistent recurrence
of the same set of logical problems.
The following examples may serve to support this thesis:
1. The problem of implication. Posed by the Megarians and Stoics
(20.05 ff.), it was resumed by the Scholastics (30.09 ff.), and again by
the mathematical logicians (41.11 ff.). Closely connected with it,
so it seems, was what the Indians called vyapli (53.20, 54.07 f.).
Perhaps more remarkable is the fact that the same results were
14
EVOLUTION OF FORMAL LOGIC
reached quite independently in different periods. Thus material
implication is defined in just the same way by Philo (20.07), Burleigh
(30.14) and Peirce (41.12 f.), in each case by means of truth-values.
Another definition is also first found among the Megarians (20.10j,
again, and this time as their main concept of implication, among the
Scholastics (30.11 f.), and is re-introduced by Lewis in 1918 (49.04).
2. The semantic paradoxes serve as a second example. Already
posed in the time of Aristotle (23.18), discussed by the Stoics
(23.20), the problem of these is found again in the Scholastics
(35.05 ff.), and forms one of the main themes in mathematical
logic (§ 48). Re-discovery of the same solutions is again in evidence
here, e.g. Russell's vicious-circle principle was already known to
Paul of Venice.
3. A third group of problems common to western logic is that of
questions about modal logic. Posed by Aristotle (§ 15), these ques-
tions were thoroughly gone into by the Scholastics (§ 33) and have
taken on a new lease of life in the latest phase of mathematical
logic (49.03).
4. We may refer again to the analysis of quantifiers : the results of
Albert of Saxony and Peirce are based on the same understanding
of the problem and run exactly parallel.
5. Similar correspondences can be noticed between Indian and
western logic. D. H. H. Ingalls has recently discovered a long
series of problems and solutions common to the two regions. Most
remarkable is the fact that Indian logic, evolving in quite different
conditions from western, and independently of it, eventually dis-
covered precisely the scholastic syllogism, and, as did western logic,
made its central problem the question of 'necessary connection'.
Still further examples could be adduced in this connection; it
seems as though there is in the history of logic a set of basic problems,
taken up again and again in spite of all differences of standpoint, and,
still more important, similarly solved again and again.
Itis not too easy to express exactly, but every reader who is a
logician will see unmistakeably the community of mind, by which
we mean the recurrent interest in certain matters, the way and
style of treating them, among all inquirers in the field of what we
comprehend within the different forms of formal logic. Read in
conjunction our texts 16.19, 22.16fL, 31.22, 33.20, 41.11 ff. There
can be no doubt that the same attitude and spirit is expressed in
them all.
E. THE PROBLEM OF PROGRESS
Closely connected with the question of the unity of logic is the
difficult problem of its progress. One thing is certain: that this
problem cannot be solved a priori by blind belief in the continuous
growth to perfection of human knowledge, but only on the basis of
15
A HISTORY OF FORMAL LOGIC
a thoroughly empirical inquiry into detail. We can only learn
whether logic has progressed in the course of its history from that
history itself. We cannot discover it by means of a philosophic
dogma.
But the problem is not easily solvable with our present historical
knowledge. One question which it involves seems indeed to be safely
answerable, but the requisites for dealing with others are still
lacking.
We can safely state the following:
1. The history of logic shows, as has already been remarked, no
linear ascending development. Consequently in the case of an advance,
it can only take place firstly, within a given period and form of
logic, and secondly, so as to raise the later forms to a higher level
than the earlier.
2. Some advance within single periods and forms of logic is
readily perceivable. We can see it best in Indian, but also in Scho-
lastic and mathematical logic. Every particular of these periods
affords a safe criterion of progress; each of them has its essential
problems, and by comparing their formulation and solution in
different logicians of the same period we can easily see that the
later writers pose the questions more sharply, apply better method
to their solution, know more laws and rules.
3. If the history of logic is considered as a whole, here too a certain
advance can be established with safety. This consists in the fact
that new problems are forthcoming in the later forms of logic.
Thus for example the highly wrought semiotic problematic of the
Scholastics is quite new in comparison with that of antiquity, and
therefore also more complete; the logical paradoxes (not the
semantic ones) of the mathematical logicians are new; so too Albert
of Saxony's problem of defining quantifiers is new. These are again
only some examples from the many possible ones.
On the other hand, the following question seems to be still
undecidable in the present state of knowledge: taking logic as a
whole, is every later form superior to all earlier ones ?
Too often this question is answered affirmatively with an eye
on mathematical logic, particularly because people compare it with
its immediate predecessor, 'classical' logic, and are struck by the
mass of laws and rules which calculation makes available in the
new form.
But 'classical' logic is by no means to be equated with the whole
of older logic; it is rather a decadent form of our science, a 'dead
period' in its evolution. Calculation, again, is certainly a useful
tool for logic, but only as facilitating new insights into logical
interconnection. It is undeniable that such insights, e.g. in the logic
of relations, have been reached by its means, and the convenience
and accuracy of this instrument are so great that no serious logician
16
EVOLUTION OF FORMAL LOGIC
can now dispense with it. But we would not go so far as to say that
calculation has at every point allowed mathematical logic to surpass
the older forms. Think for example of two-valued prepositional
logic: the essentially new features introduced by Principia Mathe-
matical are quite unimportant when we compare the scholastic
treatment.
Once again the matter reduces to our insufficient knowledge of
the earlier forms of logic. For years people spoke of a supposed
great discovery by De Morgan; then Lukasiewicz showed that his
famous law was part of the elementary doctrine of Scholasticism.
The discovery of truth-matrices was ascribed to Peirce, or even
Wittgenstein; Peirce himself found it in the Megarians. D. Ingalls
found Frege's classical definition of number in the Indian Mathu-
ranatha (17th century). And then we are all too well aware that we
know, as has been said, only fragments of Scholastic and Indian
logic, while much more awaits us in manuscripts and even in unread
printed works. The Megarian-Stoic logic, too, is lost, except for a
few poor fragments transmitted by its opponents.
Also highly relevant to the question of the continual progress of
logic throughout its history is the fact that the earlier varieties
are not simply predecessors of contemporary logic, but deal in part
with the same or similar problems though from a different stand-
point and by different methods. Now it is hard for a logician trained
in the contemporary variety of logic to think himself into another.
In other words, it is hard for him to find a criterion of comparison.
He is constantly tempted to consider valuable only what fits into
the categories of his own logic. Impressed by our technique, which is
not by itself properly logic, having only superficial knowledge of
past forms, judging from a particular standpoint, we too often risk
misunderstanding and under-rating other forms.
Even in the present state of knowledge we can be sure that
various points about the older forms still escape our comprehension.
One example is the Scholastic doctrine of supposition, which is
evidently richer in important insights and rules than the semiotic so
far developed by mathematical logic. Another is perhaps the
treatment of implication (vyapti) by the thinkers of the Navya
Nyaya. Still further examples could be given.
Again, when an unprejudiced logician reads some late-Scholastic
texts, or it may be some Stoic fragments, he cannot resist the im-
pression that their general logical level, their freedom of movement
in a very abstract domain, their exactness of formulation, while they
are equalled in our time, have by no means been excelled. The modern
mathematical logician certainly has a strong support in his calculus,
but all too frequently that same calculus leads him to dispense
with thought just where it may be most required. A conspicuous
example of this danger is provided by statements made for long
17
A HISTORY OF FORMAL LOGIC
years by mathematical logicians concerning the problem of the null
class.
These considerations tell against the thesis that logic has pro-
gressed as a whole, i.e. from variety to variety; it looks as though we
have insufficient grounds for holding it. But of course it does not at
all follow that another thesis, viz. of a purely cyclic development of
formal logic with continual recurrence of the same culminating
points, is sufficiently established.
The historian can only say; we do not know whether there is an
over-all progress in the history of logic.
§4. METHOD AND PLAN
A. HISTORY OF PROBLEMS, AND DOCUMENTATION
Conformably to the directions of the series Orbis Academicus this
work will present a documented history of problems.
We are not, therefore, presenting a material history of logic
dealing with everything that has any historical importance, but a
delineation of the history of the problematic together with the
complex of essential ideas and methods that are closely connected
with it. We only take into account those periods which have made an
essential contribution to the problematic, and among logicians those
who seem to us to rank as specially good representatives of their
period. In this connection some thinkers of outstanding importance,
Aristotle above all, Frege too, will receive much fuller treatment than
would be permissible in a material history.
The story will be told with the help of texts, and those originally
written in a foreign language have been translated into English.
This procedure, unusual in a scientific work, is justified by the consi-
deration that only a few readers could understand all the texts if
they were adduced in their original language. For even those readers
with some competence in Greek are not automatically able to under-
stand with ease a text of formal logic in that tongue. But the spe-
cialist logician will easiliy be able to find the original text by reference
to the sources.
The passages quoted will be fairly thoroughly commented where
this seems useful, for without some commentary many of them
would not be readily intelligible.
B. PLAN OF THE WORK
In itself such a history admits of being arranged according to
problems. One could consider first questions of semiotics, then
propositional ones, then those of predicate logic etc., so as to pursue
the whole history of each class of problems. E.g. the chapter on
propositional logic could begin with Aristotle, go on to the Megarian-
18
METHOD AND PLAN
Stoic theory of Xoyo<;, then to the scholastic consequentiae, to the
propositional interpretation of the Boolean calculus, to McColl, Peirce
and Frege, to chapters 2 — 5 of the Principia, finally to Lukasiewicz.
Such a method of treatment is, however, forbidden by the non-
linear evolution of logic, and above all by the fact that it takes on a
different form in every epoch. For every particular group of pro-
blems within one variety is closely connected with other complexes
of problems in the same variety. Torn from its context and ranged
with the cognate problems in another variety, it would be, not just
unintelligible, but quite misunderstood. The problem of impli-
cation provides a good example: the Scholastics put it in the
context of their theory of meaning, and their theory is not to In-
understood apart from that. Every problem considered in a given
variety of logic needs viewing in the context of the total problematic
of that variety.
It is necessary, therefore, to arrange the whole history according
to the varieties of logic. Within each we have tried to show the
connection of the various groups of problems. This has, however, not
proved to be the best course everywhere. In the discussion of
antiquity a grouping of the material according to the chronology of
logicians and schools has seemed preferable, especially because one
logician, Aristotle, has an incomparably great importance.
C. CHARACTER OF THE CONTENTS
As our knowledge of many domains is still very fragmentary, we
cannot aim at completeness. One period that is probably fairly
important, the Arabian, cannot be noticed at all. Citations from
Scholasticism are certainly only fragments. Even our knowledge of
ancient and mathematical logic is far from satisfactory. Conse-
quently this work serves rather as a survey of some aspects of the
history of logical problems than as a compilation of all that is essen-
tial to it.
What is rather aimed at is a general orientation in whatever kind
of problems, methods and notions is proper to each variety of logic,
and by that means some presentation of the general course of the
history of logic and its laws. The emphasis will be put on this course
of the problematic as a whole.
Hence we have also decided to risk a short account of Indian
logic, in spite of subjective and objective reasons to the contrary.
For this logic seems to be of great interest precisely with reference to
the laws of the whole development. At the same time it is the only
form which has developed quite independently of the others. The
chapter on Indian logic must, however, be managed differently
from the rest, not only because our knowledge of the subject is even
less sufficient than of Scholastic logic, but also because we have to
rely on translations. This chapter will be treated as a kind of appendix.
19
A HISTORY OF FORMAL LOGIC
§5. TERMINOLOGY
In order to establish a comparison between the problems and
theorems which have been formulated in different epochs and
languages, we have had to use a unified terminology in our comments.
For the most part we have taken this from the vocabulary of
contemporary formal logic. But as this vocabulary is not at all
familiar to the majority of readers, we shall here explain the most
important technical expressions.
A. TECHNICAL EXPRESSIONS
By 'expression', 'formula', 'word', 'symbol' etc. we here intend
what Morris calls the sign-vehicle, and so the material component
of the sign ; i.e. a certain quantity of ink, or bundle of sound waves. A
specially important class of expression is that of sentences, i.e.
expressions which can be characterized as true or false. It must be
stressed that a sentence, so understood, is an expression, a material
sign, and not what that signs stands for. The word 'proposition' has
been variously used, as synonymous with 'sentence' in the sense
just explained (cf. 26.03), more normally for a sentence precisely as
meaningful (Scholastics generally), sometimes with various psycho-
logical and subjective connotations (cf. the 'judgment' of the
'classical' logicians), nowadays commonly as the objective content of
a meaningful sentence (cf. the Stoic a^icopia). In our commentaries
we keep 'sentence' for the material expression, as above and use
'proposition' in the sense appropriate to the historical context and as
indicated by normal usage, which seems frequently to approximate
to that of the Scholastics.
We divide expressions into atomic and molecular (the thought is
Aristotelian, cf. 10.14 and 10.24), the former being without parts
that are themselves expressions of the given language, the latter con-
taining such parts. Molecular expressions are analysed sometimes
into subject and predicate in accordance with the tradition of Ari-
stotle and the Scholastics, sometimes into functor and argument.
The functor is the determining element, the argument the one deter-
mined ; this is also true of predicate and subject respectively, but the
other pair of terms is more general in applicability. 'And', 'not',
names of relations, are thought of as functors.
We distinguish between constant and variable expressions (again
with Aristotle, cf. 13.04), called constants and variables for short. The
former have a determinate sense, the latter only serve to mark void
places in which constants can be substituted. Thus, for example, in
'x smokes', V is a variable and 'smokes' a constant. With Frege
(42.02) we call a molecular expression which exhibits a variable a
function. Thus we speak of propositional functions, that is to say of
20
TERMINOLOGY
expressions which, if the variables that occur in them are properly
replaced by constants, become sentences (or propositions in the
Scholastic sense). lx smokes' is such a propositional function.
Among propositional functions we often mention the logical
sum or inclusive disjunction of two propositions or terms, the logical
product or conjunction, implication and equivalence. Quantifiers (cf.
44.01), 'all', 'some', 'for every x\ 'there is a y such that', are some-
times counted as functors.
Variables which can only be meaningfully replaced by propositions
we call propositional variables; such as can only be meaningfully
replaced by terms we call term-variables. Correspondingly we speak
of laws of propositional logic and term-logic. Term-logic is divided
into predicate-, class- (or classial), and relation-logic. Predicate-
logic treats of intensions, class-logic of extensions; relation-logic is
the theory of those special formal properties which belong to rela-
tions, e.g. symmetry (if Ft holds between a and 6, then it also holds
between b and a), transitivity (if B holds between a and b and be-
tween b and c, then it also holds between a and c) etc.
The general doctrine of signs we call, with W. Morris (5.01),
semioiic. This is divided into syntax (theory of the relationships
between signs), semantics (theory of the relationships between signs
and their significates), and pragmatics (theory of the relationships
between signs and those who use them). Correspondingly we speak
of syntactical, semantic and pragmatic laws and theories. In the
field of semantics we distinguish between the denotation and the
meaning or sense of a sign — which denotes the object of which it
is a sign (its reference), and means its content. (In translating
Scholastic texts we use 'signifies' for 'significaV and leave further
determination to be judged, where possible, by the context.) Thus
for example, the word 'horse' denotes a horse, but means what
makes a horse a horse, what we might call 'horseness'. We dis-
tinguish further between object-language in which the signs denote
objects that are not part of the language, and the corresponding
meta-language in which the signs denote those of the object-language.
In accordance with this terminology the word 'cat' in the sentence
'a cat is an animal' belongs to the object-language since it denotes
a non-linguistic object, but in the sentence '"cat" is a substantive'
it belongs to the meta-language, since it denotes the word 'cat' and
not a cat itself. When an expression is used as the name of another
expression that has the same form, we follow the prescription of
Frege (39.03) and write it between quotation-marks.
Finally we distinguish between logical laws and rules, as did the
Stoics (§ 22, A and B) and Scholastics (cf. the commentary on 31.13).
Laws state what is the case, rules authorize one to proceed in such
and such a way.
21
A HISTORY OF FORMAL LOGIC
B. CONCERNING MATHEMATICO-LOGICAL SYMBOLISM
In divergence from the widespread practice, which is that of the
author himself, all use of mathematico-logical symbolism has been
avoided in the commentaries on texts not of this character. In many
cases this symbolism affords easy abbreviation, and laws formulated
by its means are much easier for the specialist to read than verbally
expressed propositional functions or propositions. But two reasons
militate against its use:
1. First, objectively, it introduces an appreciable risk of misunder-
standing the text. Such a risk is present in every case of translation,
but it is particularly great when one uses a terminology with so
narrowly defined a sense as that of mathematical logic. Take, for
example, signs of implication. Those at our disposal essentially
reduce to two: 'd' and 'F'. Which of them are we to use to express
Diodorean implication? Certainly not the first, for that means
Philonian implication; but not the second either, for that would
mean that one was sure that Diodorus defined implication just like
Lewis or Buridan, which is by no means certain. Another example
is the Peano-Russellian paraphrase of Aristotelian syllogistic as it
occurs in the Principia (5.02). It is undoubtedly a misinterpretation
of Aristotle's thought, for it falsifies many laws of the syllogistic
which on another interpretation (that of Lukasiewicz) can be seen
to be correct (5.03).
Some notions not deriving from mathematical logic could indeed
be expressed in its symbolism, e.g. the Philonian implication or that
of Buridan; but to single out these for such interpretation and to
make use of verbal formulation in other cases would be to cause a
complication that is better avoided.
But of course that is not to say that no such symbolism ought
to be employed for any form of logic. For particular logicians, or
a particular form, the use of an artificial symbolism is not only
possible, but to be desired. But then every case requires a special
symbolism. What we cannot do is to create a unique symbolism
suitable for all the ideas that have been developed in the different
varieties of logic.
2. A subjective reason is provided by the limits of the work,
which aims to make allowance for the reader who is formed in the
humanities but innocent of mathematics. For such, and they are
obviously the majority, mathematico-logical symbolism would
not clarify his reading, but cloud it unnecessarily.
In these circumstances we have been at pains to use such texts
as exhibit no artificial symbolism, even in the chapter on mathe-
matical logic, so far as that is possible. Symbolic texts are of course
cited as well, and in such a way that one who wishes to acquire the
symbolic language of mathematical logic can learn the essentials
from this work. But the texts which treat of the basic problems of
22
TERMINOLOGY
logic have been chosen in such a way that they are as far as possible
intelligible without a knowledge of this symbolism.
C. TYPOGRAPHICAL CONVENTIONS
All texts are numbered decimally, the integral part giving the
paragraph in which the citation occurs, the decimals referring to a
consecutive numbering within the paragraph.
Texts are set in larger type than the commentaries, except for
formulas due to the author, which are also in larger type.
Added words are enclosed in round parentheses. Expressions in
square parentheses occur thus in the text itself. Formulas are an
exception to this: all parentheses occurring in them, together with
their contents, occur so in the original texts.
Quotation marks and italics in ancient and scholastic texts are
due to the author.
Remarks concerning textual criticism are presented in starred
footnotes.
Special points concerning the chapter on Indian logic are stated
in § 50, D.
23
PART II
The Greek Variety of Logic
§6. INTRODUCTION TO GREEK LOGIC
A. LOGICIANS IN CHRONOLOGICAL ORDER
Aristotle, the first historian of philosophy, calls Zeno of Elea the
'founder of dialectic' (6.01), but the first two men, so far as we know,
to reflect seriously on logical problems were Plato and Euclid of
Megara, both pupils of Socrates. And as Aristotle himself ascribes
to Socrates important services in the domain of logic (6.02), or
rather of methodology from which logic later developed, perhaps
Socrates should be considered to be the father of Greek logic.
Aristotle was a pupil of Plato, and his logic undoubtedly grew out
of the practice of the Platonic Academy. Aristotle's chief pupil
and long-time collaborator, Theophrastus, provides the link between
the logical thought of his master and that of the Stoa. For con-
temporaneously and parallel with Aristotelian logic there developed
that derived from Euclid, of which the first important representatives
were Megarians, Diodorus Cronus, Philo of Megara and others;
later came the Stoics, who were closely connected with the Megarians,
having Chrysippus as their most important thinker.
After the death of Chrysippus, disputes arose between the Peripa-
tetic and Megarian-Stoic schools, the latter now represented by the
Stoics alone, and syncretism became prominent. Even then logicians
were not lacking, the more important among them being apparently
the commentators on Aristotle's logical works (Alexander, Philo-
ponus), many Sceptics (especially Sextus Empiricus), these in the
3rd century B.C., and finally Boethius (5th-6th century a.d.).
The following table shows the chronological and doctrinal
connection down to Chrysippus:
Zeno of Elea, c. 464/60 b.c.
Socrates ob. 399 The ancient Sophists
Plato 428/7-348/47 Euclid of Megara c. 400
4-
Aristotle 384-322
Theophrastus ob. 287/86
4-
■Diodorus Cronus ob. 307
rJL/lUUUl US Kj
Philo o
f Megara
* Zeno of Citium
336/35-246/43
4-
Chrysippus of Soli
281/78-208/05
Peripatetic School Stoic School Megarian School
Syncretism
26
INTRODUCTION TO GREEK LOGIC
B. PERIODS
The problematic of formal logic by and large began with Aristotle.
He was undoubtedly the most fertile logician there has ever been,
in the sense that a great many logical problems were raised for the
first time in his works. Close to him in the history of ancient logic
is a group of thinkers who are nearly as important, the Megarian-
Stoic school. Aristotle lived in the 4th century B.C.; the essential
development of the Megarian-Stoic school can be thought of as
ending with the death of Ghrysippus of Soli at the end of the 3rd
century B.C. Hence in Greek antiquity there is a relatively short
period to be considered, from the second half of the 4th to the end
of the 3rd century B.C.
But that does not mean that there was no logical problematic
outside those 150 years. Even before Aristotle, a problematic
emerged in the form of the pre-Socratic and Platonic dialectic,
admittedly without ever developing into a logical theory. Again,
long after the death of Chrysippus, and right on to the end of anti-
quity, i.e. to the death of Boethius (6th century a.d.), many reflec-
tions on logical problems are to be found in the so-called Commen-
tators. This last period is not comparable in fruitfulness with that of
Aristotle and the Stoics, but we are indebted to it for various
insights worth remark.
Accordingly, from our point of view, antiquity is divided into
three main periods:
1. the preparatory period, to the time when Aristotle began to
edit his Topics.
2. the Aristotelian-Megarian-Stoic period, occupying the second
half of the 4th to the end of the 3rd century B.C.
3. the period of the Commentators, from about 200 b.c. to the
death of Boethius at the beginning of the 6th. century a.d.
The second of these periods is so outstandingly important that
it is appropriate to divide it into two sections covering respectively
Aristotle and the Megarian-Stoic school. We have then four tempo-
rally distinct sections: 1. pre-Aristotelians, 2. Aristotle and his
immediate pupils, 3. the Megarian-Stoic school, 4. Commentators.
C. STATE OF RESEARCH
The history of Greek logic is the relatively best-known period in
the development of formal logic. By contrast with the Middle
Ages and after, and to some extent with logistic too, nearly all the
surviving texts of the logicians of this age are readily available in
good modern editions, together with a whole series of scientific
treatises on their contents. In this connection there are two classes
of works:
27
THE GREEK VARIETY OF LOGIC
a) On the one hand the philologists have been busy for more
than a century with solving numerous and often difficult problems
of literary history relevant to ancient logic. Yet great as is the debt
of gratitude owed by logicians to this immense work, one cannot
pass over the fact that most philologists lack training in formal logic
and so too often overlook just the most interesting of the ancient
texts. Mostly, too, their interest centres on ontological, metaphysical,
epistemological and psychological questions, so that logic comes to
be almost always neglected. To quote only one example: logic is
allotted few pages in Polenz's two big volumes on the Stoa. Then
again editions made without a thorough logical training are often
insufficient: Kochalsky's edition of Stoic fragments may serve as
an instance.
b) On the other hand logicians too, especially since the pioneer
work of G. Vailati (1904) and A. Rustow (1908), have considered a
fair number of problems arising from these texts. Epoch-making
in this field is the article Zur Geschichle der Aussagenlogik (1935)
by J. Lukasiewicz. The same scholar has given us books on the
principle of contradiction in Aristotle and the Aristotelian (assertoric)
syllogistic. Important too are the researches of H. Scholz whose
Geschichle der Logik appeared in 1931 and who has written a number
of other studies. Each of these has formed a small school. J. Sala-
mucha investigated the concept of deduction in Aristotle (1930).
I. M. Bochenski wrote a monograph on Theophrastus (1939); his
pupils J. Stakelum (1940) and R. van den Driessche (1948) published
studies on the period of the Commentators, the former dealing with
Galen, the latter with Boethius. Boethius has also been dealt with
by K. Diirr (1952). A. Becker, a pupil of Scholz, produced an impor-
tant work on Aristotle's modal syllogisms (1933). B. Mates, influen-
ced by Lukasiewicz, has made a thorough study of Stoic logic (1953).
The state of inquiry up to now may be characterized thus:
Aristotelian studies are well opened up, though much is still missing,
e.g. discussion of the Topics; good editions of the text are also
available. We also have a very fair knowledge of Megarian-Stoic
logic, though fresh editions of the texts are desirable. Very little
work has been done on the period of the Commentators, but good
editions are mostly to hand. The pre-Aristotelian period is also very
insufficiently explored, notwithstanding the valuable studies by
A. Krokiewicz, a philologist with logical training. Especially
desirable is a thorough-going treatment of the beginnings of logic
in Plato, though admittedly such a work would meet with con-
siderable difficulties.
More exact information about the literature will be found in the
Bibliography.
28
I. THE PRECURSORS
7. THE BEGINNINGS
When Aristotle brought to a close the earliest part of his logical
work, i.e. the Topics and De Sophislicis Elenchis, he could proudly
write :
7.01 In the case of all discoveries the results of previous
labours that have been handed down from others have been
advanced bit by bit by those who have taken them on,
whereas the original discoveries generally make an advance
that is small at first though much more useful than the develop-
ment which later springs out of them. For it may be that in
everything, as the saying is, 'the first start is the main part' :
and for this reason also it is the most difficult; . . .
Of this inquiry, on the other hand, it was not the case that
part of the work had been thoroughly done before, while part
had not. Nothing existed at all.
A. TEXTS
What Aristotle says of 'this inquiry' of his seems still to hold good ;
we know of no logic, i.e. an elaborated doctrine of rules or laws,
earlier than the Topics. Certain rules of inference, however, appear
to have been consciously applied long before Aristotle by many
Greeks, without being reflectively formulated, much less axiomatized.
Aristotle himself says elsewhere that Zeno of Elea was the 'founder
of dialectic' (6.01), and it is in fact hardly possible that Zeno
formulated his famous paradoxes without being aware of the rules
he was applying. The texts ascribed to him are only to be found in
late commentators, including, however, Simplicius who was a
serious investigator; criticism casts no doubt on their authenticity.
We give some examples of his dialectic :
7.02 In the case that they (beings) are many, they must
be as many as they are, neither more nor less. But if they are
as many as they are, then they are limited (determinate). If
(however) beings are many, then they are unlimited (indeter-
minate) : since there are yet other beings between the beings
and others again between those. And thus beings are unlimited
(indeterminate).
7.03 If beings are, every one must have magnitude and volume,
and one part of it must be distinct from another . . . And so,
if they are many they must be at once small and great; small,
29
THE GREEK VARIETY OF LOGIC
since they have no magnitude, and great since they are un-
limited (indeterminate).
7.04 If there is a place, it is in something; for every being
is in something; but what is in something is also in a place.
Hence the place will itself be in a place, and so on without
end; hence there is no place.
G. Vailati stressed a text from Plato in which a similar process of
inference is used :
7.05 Socrates: And the best of the joke is, that he acknow-
ledges the truth of their opinion who believe his own opinion
to be false; for he admits that the opinions of all men are true.
Theodorus: Certainly.
Socrates: And does he not allow that his own opinion is
false, if he admits that the opinion of those who think him
false is true ?
Theodorus : Of course.
Socrates: Whereas the other side do not admit that they
speak falsely?
Theodorus: They do not.
Socrates: And he, as may be inferred from his writings,
agrees that this opinion is also true.
Theodorus: Clearly.
Socrates: Then all mankind, beginning with Protagoras, will
contend, or rather, I should say that he will allow, when he
concedes that his adversary has a true opinion, Protagoras,
I say, will himself allow that neither a dog nor any ordinary
man is the measure of anything which he has not learned - am
I not right?
Theodorus: Yes.
The big fragment of Gorgias (7.06) also contains something
similar, but this is so evidently composed in the technical terminology
of the Stoics and betrays so highly developed a technique of logical
thought that we cannot ascribe it to the Sophists, nor even to
Aristotle. It is, however, possible that the young Aristotle did indeed
formulate the famous proof of the necessity of philosophy in the way
which it ascribed to him. This proof is transmitted to us in the follow-
ing three passages among others:
7.07 There are cases in which, whatever view we adopt, we
can refute on that ground a proposition under consideration.
So for instance, if someone was to say that it is needless to
philosophize : since the enquiry whether one needs to philo-
30
PRECURSORS
sophize or not involves philosophizing, as he (Aristotle) has
himself said in the Protreplicus, and since the exercise of
a philosophical pursuit is itself to philosophize. In showing
that both positions characterize the man in every case, we
shall refute the thesis propounded. In this case one can rest
one's proof on both views.
7.08 Or as Aristotle says in the work entitled Protreplicus
in which he encourages the young to philosophize. For he
says: if one must philosophize, then one must philosophize; if
one does not have to philosophize, one must still philosophize.
So in any case one has to philosophize.
7.09 Of the same kind is the Aristotelian dictum in the
Protreplicus: whether one has to philosophize or not, one
must philosophize. But either one must philosophize or not;
hence one must in any case philosophize.
B. SIGNIFICANCE
All the texts adduced above spring from the milieu of 'dialectic'.
This word that is later given so many meanings and is so mis-used
originally had the same meaning as our 'discussion'. It is a matter
of disputation between two speakers or writers. That is probably the
reason why most of the rules of inference used here - termed, as it
seems, iogoi' - lead to negative conclusions: the purpose was to
refute something, to show that the assertion propounded by the
opponent is false.
This suggests the conjecture that these logoi belong to the field
of propositional logic, that is to say that it is here a matter of
logical relations between propositions as wholes without any ana-
lysis of their structure. And in fact the pre-Aristotelian logoi were
often so understood. However, this interpretation seems untenable:
Aristotle himself was aware of the very abstract laws of propositional
logic only exceptionally and at the end of his scientific career; so
much the less ought we to ascribe this - Megarian-Stoic - manner of
thought to the pre-Aristotelians. We have rather to do with certain
specifications of general rules of propositional logic. Thus these
dialecticians were not thinking of, for example, the abstract scheme
of propositional logic corresponding to modus ponendo ponens:
7.101 If p, then q; but p ; therefore q:
but rather of the more special law
7.102 If A belongs to x, then B also belongs to x; but .4.
belongs to x; therefore B also belongs to x.
31
THE GREEK VARIETY OF LOGIC
We purposely omit quantifiers here, since while such were necessa-
rily present to the thought obscurely, at this level there can be no
question of a conscious acceptance of such logical apparatus.
We note further, that at the level of pre-Aristotelian dialectic, it is
always a matter of rules not of laws; they are principles stating how
one should proceed, not laws, which describe an objective state of
affairs. That does not mean of course that the dialecticians were in
any way conscious of the distinction between the two; but from our
point of view, what they used were rules.
This said, we can interpret as follows the several logoi previously
adduced. For each we give the logical sentence corresponding to the
rule of inference which it employs.
Zeno quoted by Simplicius (7.02, 03, 04) :
7.021 If A belongs to x then B and C also belong to x; but
B and C do not belong to x; therefore neither does A belong
to x.
7.022 Suppose that if A belongs to x, B also belongs to x
and if B belongs to x, C also belongs to x, then if A belongs to
x, C also belongs to x.
Plato in the Theaeletus (7.05) :
7.051 If A belongs to x then A does not belong to x;
therefore A does not belong to x.
Closer examination of that last item shows that it is much more
complex and belongs to the realm of metalogic. Plato's thought
proceeds after this fashion: the proposition propounded by Protago-
ras means: for every x, if x says 'p', then p. Let us abbreviate that
by *S\ Now there is some (at least one) x who says that S is not the
case. Therefore S is not the case. Therefore if S, then not S. From
which it follows in accordance with 7.051, that S is not the case.
While Plato certainly did not expressly draw this conclusion, he
evidently intended it.
Aristotle quoted by Alexander (7.07) :
7.071 Suppose that if A belongs to x, A belongs to x, and
if A does not belong to x, A belongs to a?, then A belongs to x.
The anonymous scholiast has a fuller formula (7.09) :
7.091 If A belongs to aj, then A belongs to x; if A does not
belong to x, then A belongs to x; either A belongs to x or A
does not belong to x; therefore A belongs to x,
but whether it actually occurred in Aristotle may be doubted.
Possibly the Protreplicus contained merely the simple formula,
transmitted by Lactantius :
32
PRECURSORS
7.092 If A does not belong to x, then A belongs to x;
therefore A belongs to x.
A series of similar formulae underlie the processes to be found in
the great Gorgias-fragment (7.06), but these appear to be so markedly
interpreted in the light of Stoic logic that we have no guarantee of
anything genuinely due to the sophist himself.
§ 8. PLATO
While Plato, in respect of many rules used in his dialectic,
belongs to the same period as Zeno (as too does the youthful Ari-
stotle), he begins something essentially new in our field, and that
from several points of view.
A. CONCEPT OF LOGIC
In the first place Plato rendered the immortal service of being the
first to grasp and formulate a clear idea of logic. The relevant text
occurs in the Timaeus and runs :
8.01 God invented and gave us sight to the end that we
might behold the courses of intelligence in the heaven, and
apply them to the courses of our own intelligence which are
akin to them, the unperturbed to the perturbed; and that we,
learning them and partaking of the natural truth of reason,
might imitate the absolutely unerring courses of God and
regulate our own vagaries.
Such a conception of logic was, however, only possible for Plato,
because he was, as it seems, the originator of another quite original
idea, namely that of universally necessary laws (granting that he
depended in this on the logos-doctrine of Heracleitus and other
earlier thinkers). The concept of such laws is closely connected with
Plato's theory of ideas, which itself developed through reflection on
Geometry as it then existed. The whole post-Platonic western
tradition is so penetrated with these ideas, that it is not easy for a
westerner to grasp their enormous significance. Evidently no formal
logic was possible without the notion of universally valid law. From
this point of view the importance of Plato for the history of logic
can best be seen when we consider the development of the science in
India, i.e. in a culture which had to create logic without a Plato. One
can see in the history of Indian logic that it took hundreds of years
to accomplish what was done in Greece in a generation thanks to
the elan of Plato's genius, namely to rise to the standpoint of uni-
versal validity.
33
THE GREEK VARIETY OF LOGIC
We cannot here expound Plato's doctrine of ideas, as it belongs to
ontology and metaphysics, and is further beset with difficult pro-
blems of literary history.
B. APPROACHES TO LOGICAL FORMULAE
Plato tried throughout his life to realize the ideal of a logic as laid
down above, but without success. The tollowing extracts from his
dialectic, in which he makes a laboured approach to quite simple
laws, show how difficult he found it to solve logical questions that
seem elementary to us.
8.02 Socrates: Then I shall proceed to add, that if the
temperate soul is the good soul, the soul which is in the
opposite condition, that is, the foolish and intemperate, is
the bad soul. - Very true. - And will not the temperate man
do what is proper, both in relation to the gods and to men; -
for he would not be temperate if he did not? - Certainly he
will do what is proper.
8.03 Socrates: Tell me, then, - Is not that which is pious
necessarily just?
Euthyphro: Yes.
Socrates: And is, then, all which is just pious? or, is that
which is pious all just, but that which is just, only in part and
not all, pious?
Euthyphro: I do not understand you, Socrates.
8.04 When you asked me, I certainly did say that the
courageous are the confident; but I was never asked whether
the confident are the courageous; if you had asked me, I
should have answered 'Not all of them :' and what I did answer
you have not proved to be false, although you proceeded to
show that those who have knowledge are more courageous
than they were before they had knowledge, and more coura-
geous than others who have no knowledge, and were then led
on to think that courage is the same as wisdom. But in this
way of arguing you might come to imagine that strength is
wisdom. You might begin by asking whether the strong are
able, and I should say 'Yes'; and then whether those who
know how to wrestle are not more able to wrestle than those
who do not know how to wrestle, and more able after than
before they had learned, and I should assent. And when I had
admitted this, you might use my admissions in such a way as
to prove that upon my view wisdom is strength; whereas
in that case I should not have admitted, any more than in the
34
PRECURSORS
other, that the able are strong, although I have admitted that
the strong are able. For there is a difference between ability
and strength; the former is given by knowledge as well as by
madness or rage, but strength comes from nature and a
healthy state of the body. And in like manner J say of confi-
dence and courage, that they are not the same; and I argue
that the courageous are confident, but not all the confident
courageous. For confidence may be given to men by art, and
also, like ability, by madness and rage; but courage comes to
them from nature and the healthy state of the soul.
In the first of these texts is involved the (false) thesis: Suppose, if
A belongs to x, B also belongs to x, then: if A does not belong to x,
then B does not belong to x. The second shows the difficulties found
concerning the convertibility of universal affirmative sentences: viz.
whether 'all B is A' follows from 'all A is B ' . The third text shows
still more clearly how hard Plato felt these questions to be; it further
has the great interest that, to show the invalidity of the fore-
going rule of conversion, he betakes himself to complicated extra-
logical discussions - about bodily strength, for instance.
C. DIAERESIS
Yet Plato's approximations were not without fruit. He seems to
have been the first to progress from a negative dialectic to the con-
cept of positive proof; for him the aim of dialectic is not to refute
the opinions of opponents but positive 'definition of the essence'. In
this he definitely directed attention to the logic of predicates, which
is probably the cause of Aristotelian logic taking the form it did.
The chief goal which Plato set himself was to discover essences, i.e. to
find statements which between them define what an object is. For
this he found a special method - the first logical, consciously ela-
borated inferential procedure known to us - namely his famous
'hunt' for the definition by division (Sioctpeais). How thoroughly
conscious he was of not only using such a method but of endeavour-
ing to give it the clearest possible formulation, we see in the cele-
brated text of the Sophist in which the method, before being practis-
ed, is applied in an easy example:
8.05 Stranger: Meanwhile you and I will begin together and
enquire into the nature of the Sophist, first of the three : I
should like you to make out what he is and bring him to light
in a discussion; for at present we are only agreed about the
name, but of the thing to which we both apply the name
possibly you have one notion and I another; whereas we ought
always to come to an understanding about the thing itself in
35
THE GREEK VARIETY OF LOGIC
terms of a definition, and not merely about the name minus the
definition. Now the tribe of Sophists which we are investi-
gation is not easily caught or defined ; and the world has long
ago agreed, that if great subjects are to be adequately treated,
they must be studied in the lesser and easier instances of them
before we proceed to the greatest of all. And as I know that the
tribe of Sophists is troublesome and hard to be caught, I
should recommend that we practise beforehand the method
which is to be applied to him on some simple and smaller thing,
unless you can suggest a better way.
Theaetetus: Indeed I cannot.
Stranger: Then suppose that we work out some lesser
example which will be a pattern of the greater?
Theaetetus : Good.
Stranger: What is there which is well known and not great,
and is yet as susceptible of definition as any larger thing? Shall
I say an angler? He is familiar to all of us, and not a very
interesting or important person.
Theaetetus : He is not.
Stranger: Yet I suspect that he will furnish us with the sort
of definition and line of enquiry which we want.
Theaetetus: Very good.
Stranger: Let us begin by asking whether he is a man having
art or not having art, but some other power.
Theaetetus: He is clearly a man of art.
Stranger: And of arts there are two kinds?
Stranger: Seeing, then, that all arts are either acquisitive or
creative, in which class shall we place the art of the angler?
Theaetetus : Clearly in the acquisitive class.
Stranger: And the acquisitive may be subdivided into two
parts: there is exchange, which is voluntary and is effected by
gifts, hire, purchase; and the other part of acquisitive, which
takes by force of word or deed, may be termed conquest?
Theaetetus: That is implied in what has been said.
Stranger: And may not conquest be again subdivided?
Theaetetus : How?
Stranger: Open force may be called fighting, and secret
force may have the general name of hunting?
Theaetetus : Yes.
Stranger: And there is no reason why the art of hunting
should not be further divided.
36
PRECURSORS
Theaetetus : How would you make the division?
Stranger: Into the hunting of living and of lifeless prey.
Theaetetus :Yes, if both kinds exist.
Stranger: Of course they exist; but the hunting after life-
less things having no special name, except some sorts of
diving, and other small matters, may be omitted; the hunting
after living things may be called animal hunting.
Theaetetus : Yes.
Stranger: And animal hunting may be truly said to have two
divisions, land-animal hunting, which has many kinds and
names, and water-animal hunting, or the hunting after ani-
mals who swim?
Theaetetus: True.
Stranger: And of swimming animals, one class lives on the
wing and the other in the water?
Theaetetus : Certainly.
Stranger: Fowling is the general term under which the
hunting of all birds is included.
Theaetetus : True.
Stranger: The hunting of animals who live in the water has
the general name of fishing.
Theaetetus : Yes.
Stranger: And this sort of hunting may be further divided
also into two principal kinds?
Theaetetus : What are they?
Stranger: There is one kind which takes them in nets, another
which takes them by a blow.
Theaetetus: What do you mean, and how do you distinguish
them ?
Stranger : As to the first kind - all that surrounds and encloses
anything to prevent egress, may be rightly called an enclosure.
Theaetetus: Very true.
Stranger: For which reason twig baskets, casting-nets,
nooses, creels, and the like may all be termed 'enclosures'?
Theaetetus: True.
Stranger: And therefore this first kind of capture may be
called by us capture with enclosures, or something of that sort ?
Theaetetus: Yes.
Stranger: The other kind, which is practised by a blow with
hooks and three-pronged spears, when summed up under one
name, may be called striking, unless you, Theaetetus, can
find some better name ?
37
THE GREEK VARIETY OF LOGIC
Theaetetus : Never mind the name - what you suggest will
do very well.
Stranger: There is one mode of striking, which is done at
night, and by the light of a fire, and is by the hunters them-
selves called firing, or spearing by firelight.
Theaetetus: True.
Stranger: And the fishing by day is called by the general
name of barbing, because the spears, too, are barbed at the
point.
Theaetetus: Yes, that is the term.
Stranger: Of this barb-fishing, that which strikes the fish
who is below from above is called spearing, because this is
the way in which the three-pronged spears are mostly used.
Theaetetus : Yes, it is often called so.
Stranger: Then now there is only one kind remaining.
Theaetetus : What is that ?
Stranger: When a hook is used, and the fish is not struck in
any chance part of his body, as he is with the spear, but only
about the head and mouth, and is then drawn out from below
upwards with reeds and rods: - What is the right name of
that mode of fishing, Theaetetus?
Theaetetus : I suspect that we have now discovered the object
of our search.
Stranger: Then now you and I have come to an under-
standing not only about the name of the angler's art, but
about the definition of the thing itself. One half of all art was
acquisitive - half of the acquisitive art was conquest or
taking by force, half of this was hunting, and half of hunting
was hunting animals, half of this was hunting water animals -
of this again, the under half was fishing, half of fishing was
striking; a part of striking was fishing with a barb, and one
half of this again, being the kind which strikes with a hook
and draws the fish from below upwards, is the art which we
have been seeking, and which from the nature of the opera-
tion is denoted angling or drawing up.
Theaetetus: The result has been quite satisfactorily brought
out.
The process is evidently not conclusive: as Aristotle has forcibly
shown (8.06), it involves a succession of assertions, not a proof; it
may be helpful as a method, but it is not formal logic.
Formal logic is reserved for Aristotle. But a close examination of
the contents of his logical works assures us that everything contained
38
PRECURSORS
in the Organon is conditioned in one way or another by the practice
of Platonism. The Topics is probably only a conscious elaboration of
the numerous logoi current in the Academy; even the Analytics,
invention of Aristotle's own as it was, is evidently based on
'division', which it improved and raised to the level of a genuine
logical process. That is the second great service which Plato
rendered to formal logic: his thought made possible the emergence
of the science with Aristotle.
39
II. ARISTOTLE
§9. THE WORK OF ARISTOTLE AND THE PRO-
BLEMS OF ITS LITERARY HISTORY
The surviving logical works of Aristotle set many difficult pro-
blems of literary history which as yet are only partly solved. They are
of outstanding importance for the history of the problems of logic,
since within the short span of Aristotle's life formal logic seems to
have made more progress than in any other epoch. It is no exagge-
ration to say that Aristotle has a unique place in the history of
logic in that 1. he was the first formal logician, 2. he developed formal
logic in at least two (perhaps three) different forms, 3. he consciously
elaborated some parts of it in a remarkably complete way. Further-
more, he exercised a decisive influence on the history of logic for
more than two thousand years, and even today much of the doctrine
is traceable back to him. It follows that an adequate understanding
of the development of his logical thought is of extreme importance
for an appreciation of the history of logical problems in general, and
particularly of course for western logic.
A. WORKS
The surviving works of the Stagirite were set in order and edited
by Andronicus of Rhodes in the first century b.c. The resulting
Corpus Aristotelicum contains, as to logical works, first and fore-
most what was later called the Organon, comprising:
1. The Categories,
2. About Propositions (properly: About Interpretation; we shall
use the title Hermeneia),
3. The Prior Analytics, two books: A and B,
4. The Posterior Analytics, two books : A and B,
5. The Topics, eight books: A,B, T, A, E, Z, H, 0,
6. The Sophistic Refutations, one book.
Besides these, the whole fourth book (T) of the so-called Meta-
physics is concerned with logical problems, while other works, e.g. the
Rhetoric and Poetics contain occasional points of logic.
B. PROBLEMS
The most important problems concerning the Organon are the
following :
1. Authenticity
In the past the genuineness of all Aristotle's logical writings has
often been doubted. Today, apart from isolated passages and
perhaps individual chapters, the Categories alone is seriously con-
sidered to be spurious. The doubt about the genuineness of the
40
A R I S T O T L E
Hermeneia seems not convincing. The remaining works rank by and
large as genuine. *
2. Character
Should we view the logical works of Aristotle as methodically
constructed and systematic treatises? Researches made hitherto
allow us to suppose this only for some parts of the Organon. The
Hermeneia and Topics enjoy the relatively greatest unity. The
Prior Analytics are evidently composed of several strata, while the
Posterior Analytics are mainly rather a collection of notes for lectures
than a systematic work. But even in those parts of the Organon that
are systematically constructed later additions are to be found here
and there.
3. Chronology
The Organon, arranged as we have it, is constructed on a syste-
matic principle: the Categories treats of terms, the Hermeneia of
propositions, the remaining works of inference: thus the Prior
Analytics treats of syllogisms in general, the three other works
successively of apodeictic (scientific), dialectical and sophistical
syllogisms. For this systematization Andronicus found support in
the very text of the Organon; e.g. at the beginning of the Prior
Analytics it is said (9.02) that the syllogism consists of propositions
(npOTCcaic,) , these of terms (6po«;). In the Topics (9.03) and also in
the Prior Analytics (9.04) the syllogism is analysed in just that way.
At the end of the Sophistic Refutations occurs the sentence already
cited (7.01), which appears to indicate that this work is the latest
of Aristotle's logical works.
It is also not impossible that at the end of his life Aristotle himself
drafted an arrangement of his logic and accordingly ordered his
notes and treatises somewhat as follows. But of course this late
systematization has, to our present knowledge, little to do with the
actual development of this logic.
We have no extrinsic criteria to help us establish the chronolo-
gical sequence of the different parts of the Organon. On the other
hand their content affords some assistance, as will now be briefly
explained.**
* The thesis of Josef Ziircher (9.01) that nearly all the formal logic in the
Organon is due not to Aristotle but to his pupil Theophrastus, is not worth
serious consideration.
* * Chr. Brandis opened up the great matter of the literary problems of the
Organon in his paper Vber die Reihenfolge des aristotelischen Organons (9.05); the
well-known work of W. Jager (9.06) contributed important insights; its basic
pre-suppositions were applied to the Organon by F. Solmsen (9.07). Solmsen's
opinions were submitted to a thorough criticism by Sir W. D. Ross (9.08) with
an adverse result in some cases. Important contributions to the chronology of
the Organon are to be found in A. Becker (9.09) and J. Lukasiewicz ^9.10).
41
THE GREEK VARIETY OF LOGIC
a. Chronological criteria
aa) A first criterion to determine the relative date of origin is
afforded by the fact that the syllogism in the sense of the Prior
Analytics (we shall call it the 'analytical syllogism') is completely
absent from several parts of the Organon. But it is one of the most
important discoveries, and it can hardly be imagined that Aristotle
would have failed to make use of it, once he had made it. We
conclude that works in which there are no analytic syllogisms are
earlier than those in which they occur.
bb) In some parts of the Organon we find variables (viz. the letters
A, B, r, etc.), in others not. But variables are another epoch-
making discovery in the domain of logic, and the degree to which it
impressed Aristotle can be seen in the places where he uses and
abuses them to the point of tediousness. Now there are some works
where variables would be very useful but where they do not occur.
We suppose that these works are earlier than those where they do
occur.
cc) The third criterion - afforded by the technical level of the
thought - cannot, unlike the first two, be formulated simply, but is
apparent to every experienced logician at his first perusal of a text.
From this point of view there are big differences between the
various passages of the Organon: in some we find ourselves at a still
very primitive level, reminiscent of pre-Socratic logic, while in
others Aristotle shows himself to be the master of a strictly formal
and very pure logical technique. One aspect of this progress appears
in the constantly developing analysis of statements: at first this
is accomplished by means of the simple subject-predicate schema
(S — P), then quantifiers occur (lP belongs [does not belong] to all,
to none, to some S'), finally we meet a subtle formula that reminds
us of the modern formal implication : 'All that belongs to S, belongs
also to P\ This criterion can be formulated as follows: the higher
and more formal the technique of analysis and proof, so much the
later is the work.
dd) Modal logic corresponds much better with Aristotle's own
philosophy (which contains the doctrine of act and potency as an
essential feature) than does purely assertoric logic in which the
distinction between act and potency obtains no expression. Asser-
toric logic fits much better with the Platonism to which Aristotle
subscribed in his youth. Accordingly we may view those writings
and chapters containing modal logic as having been composed later.
ee) Some of these criteria can be further sharpened. Thus we can
trace some development in the theory of analytic syllogism. Again,
Aristotle seems to have used letters at first as mere abbreviations
for words and only later as genuine variables. Finally one can detect
a not insignificant progress in the structure of modal logic.
It may certainly be doubted whether any one of these criteria is
42
ARISTOTLE
of value by itself for establishing the chronology. But when all, or
at least several of them point in the same direction, the resulting
sequence seems to enjoy as high a degree of probability as is ever
possible in the historical sciences.
b. Chronological list
The application of these criteria enable us to draw up the following
chronological list of Aristotle's logical writings:
aa) The Topics (together with the Categories if this is to be accep-
ted as genuine) undoubtedly comes at the start. There is to be found
in it no trace of the analytic syllogism, no variables, no modal logic,
and the technical level of the thought is relatively low. While the
Sophistic Refutations simply forms the last book of the Topics, it
appears to have been composed a little later. Book F of the Meta-
physics probably belongs to the same period. The Topics and the
Refutations together contain Aristotle's first logic. The remark at
the end of the Sophistic Refutations about the 'whole' of logic
refers to that elaboration.
bb) The Hermeneia and - perhaps - book B of the Posterior
Analytics form a kind of transitional stage: the syllogistic can be
seen emerging. In the Hermeneia we hear nothing of syllogism and
there are no variables. Both, but evidently only in an early stage
of development, occur in Posterior Analytics book B. The technical
level of thought is much higher than in the Topics. The Hermeneia
also contains a doctrine of modality, which is, however, quite
primitive compared with that in the Prior Analytics.
cc) Book A of the Prior Analytics, with the exception of chapters
8-22, contains Aristotle's second logic, a fully developed assertoric
syllogistic. He is by now in possession of a clear idea of analytic
syllogism, uses variables with sureness, and moves freely at a rela-
tively high technical level. The analysis of propositions has been
deepened. Missing, as yet, are modal logic, and reflective considera-
tion of the syllogistic system. Perhaps book A of the Posterior
Analytics may be ascribed to the same period. Solmsen made this
the first of all the analytic books, but W. D. Ross's arguments
against this seem convincing. (The latter holds that book B of the
Posterior Analytics is also later than the Prior.)
dd) Finally we may ascribe to a still later period chapters 8-22 of
book A, which contains the modal syllogistic logic, and book B of
the Prior Analytics. These can be said to contain Aristotle's third
logic, which differs less from the second than does the second from
the first. We find here a developed modal logic, marred admittedly
by many incompletenesses and evidently not finished, and also
penetrating remarks, partly metalogical, about the system of
43
THE GREEK VARIETY OF LOGIC
syllogistic. In them Aristotle offers us insights into formal logic of
remarkable subtlety and acuteness. He states too some theorems
of propositional logic with the aid of propositional variables.
Of course there can be no question of absolute certainty in answer-
ing the chronological questions, especially as the text is corrupt in
many places or sprinkled with bits from other periods. It is only
certain that the Topics and Sophistic Refutations contain a different
and earlier logic than the Analytics, and that the Hermeneia exhi-
bits an intermediate stage. For the rest we have well-founded
hypotheses which can lay claim at least to great probability.
In accordance with these hypotheses we shall speak of three
logics of Aristotle.
§10. CONCEPT OF LOGIC. SEMIOTIC
A. NAME AND PLACE OF LOGIC
Aristotle has no special technical name for logic: what we now
call 'logical' he calls 'analytic' (ocvocXutlxo^: 10.01) or 'following
from the premisses' (ex tgW xeiuivcov: 10.02), while the expression
'logical' (Xoyix6<;) means the same as our 'probable' (10.03) or
again 'epistemological'.
10.04 Of propositions and problems there are - to com-
prehend the matter in outline - three divisions : for some are
ethical propositions, some are on natural philosophy, while
some are logical. . . such as this are logical, e.g. 'Is the know-
ledge of opposites the same or not?'
The question whether logic is a part of philosophy or its instrument
(opyavov) - and hence an art - is nowhere raised by Aristotle in the
extant works.
B. THE SUBJECT-MATTER OF LOGIC
Yet Aristotle knew well enough what he demanded of logic.
That appears from the model statements of the subject-matter of
his logical treatises. For instance he says in the Topics:
10.05 First then we must say what reasoning is, and what
its varieties are, in order to grasp dialectical reasoning: for
this is the object of our search in the treatise before us. Now
reasoning is an argument in which, certain things being laid
down, something other than these necessarily comes about
through them. It is a 'demonstration' when the premisses
from which the reasoning starts are true and primary . . .
44
ARISTOTLE
reasoning, on the other hand, is 'dialectical', if it reasons
from opinions that are generally accepted. . . . Again, reason-
ing is 'contentious' if it starts from opinions that seem to be
generally accepted, but are not really such, or again if it
merely seems to reason from opinions that are or seem to be
generally accepted.
Compare the following text from the Prior Analytics:
10.06 After these distinctions we now state by what means,
when, and how every syllogism is produced; subsequently
we must speak of demonstration. Syllogism should be discuss-
ed before demonstration, because syllogism is the more
general: demonstration is a sort of syllogism, but not every
syllogism is a demonstration.
The thought is perfectly clear: Aristotle is looking for relations
of dependence which authorize necessary inference, and in that
connection makes a sharp distinction between the validity of this
relation and the kind of premisses, or their truth. The text contains
what is historically the first formulation of the concept of a formal
logic, universally valid and independent of subject-matter.
Accordingly it is syllogism which is the subject of logic. This is
a form of speech (Xoyo^) consisting of premisses (nporuikaeu;)
themselves composed of terms (5poi). 'Premiss' and 'term' are thus
defined by Aristotle:
10.07 A premiss is a form of speech which affirms or
denies something of something. ... A term I call that into
which the premiss is resolved, that is to say what is predicated
and that of which it is predicated by means of the addition
of being or not being.
What emerges from that text is the complete neutrality of the
technical expressions 'term', 'premiss', 'syllogism', relative to any
philosophical interpretation. For the premiss consists of terms, the
syllogism of premisses, and premisses are logoi, which can equally
well mean utterances or thoughts or objective contents, so that the
way is open to a formalist, psychological or objectivist interpreta-
tion. All these interpretations are permissible in regard to Aristo-
telian logic; the purely logical system excludes none of them. Guided
by his original intuition the founder of formal logic so chose his
terminology as to rise above the clash of interpretations to the level
of pure logic.
However if one considers Aristotelian logic in its entirety, it is
easy to see that this neutrality is not the result of a lack of interest in
problems of interpretation, but is on the contrary an abstraction
45
THE GREEK VARIETY OF LOGIC
from a complex semiotic doctrine. In some places Aristotle seems to
plead for a psychological type of theory, as when, for example, he
says:
10.08 All syllogism and therefore a fortiori demonstration,
is addressed not to outward speech but to that within the
soul.
At the same time it must be said that he attaches great importance
to the 'outward speech', since he elaborates a well-developed
theory of logical syntax and many points of semantic interest.
All this teaching, which is next to be considered, warrants the con-
clusion that the practice of Aristotelian logic was undoubtedly to
regard meaningful words as its subject-matter.
C. SYNTAX
Aristotle is the founder of logical syntax, following here some
hints of the Sophists and Plato. He sketched the first attempt known
to us at a system of syntactical categories. For we find in the Her-
meneia an explicit division of the parts of speech into atomic (nouns
and verbs) and molecular (sentences).
10.09 By a noun we mean a sound significant by con-
vention, which has no reference to time, and of which no
part is significant apart from the rest.
10.10 A verb is that which, in addition to its proper
meaning, carries with it the notion of time. No part of it has
any independent meaning, and it is a sign of something said
of something else.
This theory is further supplemented by discussions of cases and
inflexions of words, and by considerations about negated nouns and
verbs.
10.11 A sentence is a significant portion of speech, some
parts of which have an independent meaning, that is to say,
as an utterance (^olgiq) , though not as the expression of any
positive judgement (xaTowpaais).
10.12 The first unified declarative sentence is the affirma-
tion; the next, the denial. All other sentences are unified
by combination.
Anticipating the further explanations, we may summarize the
whole scheme of syntactical categories as presented in the Herme-
neia, after this fashion :
46
A FU S T O T L E
conven-
tionally
significant
sound
(10.13)
atomic
(10.14)
mole-
cular
Xoyoc,
(10.24)
noun
(6vo[i.a) in
broad
sense
other (poLGZic,
statement
qltzo^olvgic,
(10.25)
noun in
strict
sense (10.10)
simple
composite
(10.17)
individual
miversal
10.18]
negated noun (10.19)
cases of noun (10.20)
verb p7)[i.a (10.21)
negated verb (10.22)
inflexions of verbs (10.23)
affirmative
xaxacpac!.?
negative
OLTCOyOLGIC,
atomic
(10.26)
with singu-
lar subject
with
universal
subject
taken not taken
universally universallv
(10.27)
molecular (10.26)
other Xoyoi, (also called cpfxaeic,) (10.26)
This schema underlies the whole development of logical syntax,
and semantics too, until the rise of mathematical logic. Only this
last will introduce anything essentially new: the attempt to treat
syntactical categories by means of an artificial language. Aristotle
on the other hand, and with him the Stoic and scholastic traditions,
sought to grasp the syntactical structure of ordinary language.
D. SEMANTICS
The texts previously cited already contain points belonging to
the domain of semantics. The general principle is thus formulated by
Aristotle:
47
THE GREEK VARIETY OF LOGIC
10.28 Spoken words are the symbols of mental experience
and written words are the symbols of spoken words.
It follows that thoughts are themselves symbols of things.
Aristotle lays great stress on the parallelism between things,
thoughts and symbols, and correspondingly develops two important
semiotic theories:
10.29 Things are said to be named 'equivocally' when,
though they have a common name, the definitions correspond-
ing with the name differs for each. Thus a real man and a
figure in a picture can both lay claim to the name 'animal'. . . .
Things are said to be named 'univocally' which have both
the name and the definition answering to the name in com-
mon. A man and an ox are both 'animal' ....
Things are said to be named 'derivatively', which derive
their name from some other name, but differ from it in
termination. Thus the grammarian derives his name from the
word 'grammar' and the courageous man from the word
'courage'.
Equivocity must be excluded from demonstrations, since it
leads to fallacies (10.30). Elsewhere Aristotle distinguishes various
kinds of equivocity.
10.31 The good, therefore, is not some common element
answering to one Idea. But what then do we mean by the
good ? It is surely not like the things that only chance to have
the same name. Are goods one, then, by being derived from
one good or by all contributing to one good, or are they
rather one by analogy? Certainly as sight is in the body, so
is reason in the soul, and so on in other cases.
This division can be presented schematically as follows:
equivocal
expressions
in strict sense
dcTco tux"/)*;
(accidentally equivocal
in broad sense
(systematically equivocal)
from one (deep* hoc,)
to one (npbq lv)
by proportion
(xoct' avaXoytav)
48
ARISTOTLE
In the Metaphysics and Hermeneia we find a clear sern ioric
theory of truth :
10.32 For falsity and truth are not in things - it is not as if
the good were true and the bad were in itself false - but in
thought.
10.33 As there are in the mind thoughts which do not
involve truth or falsity, and also those which must be either
true or false, so it is in speech. For truth and falsity imply com-
bination and separation. Nouns and verbs, provided nothing
is added, are like thoughts without combination or separation.
10.34 Yet not every sentence states something, but only
those in which there is truth or falsity, and not all are of
that kind. Thus a prayer is a sentence, but is neither true
nor false . . . the present theory is concerned with such sen-
tences as are statements (#7co<pocvTix&s X6yo<;).
Aristotle also constructs a definition of truth by means of equi-
valences:
10.35 If it is true to say that a thing is white, it must
necessarily be white; if the reverse proposition is true, it
will of necessity not be white. Again, if it is white, the proposi-
tion stating that it is white was true; if it is not white, the
proposition to the opposite effect was true. And if it is not
white , the man who states that it is is making a false statement ;
and if the man who states that it is white is making a false
statement, it follows that it is not white.
11. THE TOPICS
A. SUBJECT AND PURPOSE
The Topics contains Aristotle's first logic, and so the first attempt
at a systematic presentation of our science. We cannot here do
more than glance over the mass of rules contained in this work,
giving only its purpose, a discussion of the analysis of statements
as made by Aristotle in this early work, and a brief review of his
teaching about fallacies. The most important of the formal rules
and laws of inference occurring here, continued, as it seems, to be
recognized as valid in the later works, and they will therefore be
considered in the section on the non-analytical formulae (§ 16).
49
THE GREEK VARIETY OF LOGIC
11.01 Next in order after the foregoing, we must say for
how many and for what purposes the treatise is useful. They
are three - intellectual training, casual encounters, and the
philosophical sciences. That it is useful as a training is obvious
on the face of it. The possession of a plan of inquiry will
enable us more easily to argue about the subject proposed.
For purposes of casual encounters, it is useful because when
we have counted up the opinions held by most people, we shall
meet them on the ground not of other people's convictions
but of their own, while we shift the ground of any argument
that they appear to us to state unsoundly. For the study of
the philosophical sciences it is useful, because the ability to
raise searching difficulties on both sides of a subject will make
us detect more easily the truth and arror about the several
points that arise. It has a further use in relation to the
ultimate bases of the principles used in the several sciences.
For it is impossible to discuss them at all from the principles
proper to the particular science in hand, seeing that the prin-
ciples are the prius of everything else : it is through the
opinions generally held on the particular points that these
have to be discussed, and this task belongs properly, or most
appropriately, to dialectic: for dialectic is a process of criti-
cism wherein lies the path to the principles of all inquiries.
The logic thus delineated treats of propositions and problems,
described as follows:
11.02 The materials with which arguments start are equal
in number, and are identical, with the subjects on which
reasonings take place. For arguments start with 'propositions',
while the subjects on which reasonings take place are 'pro-
blems'.
11.03 The difference between a problem and a proposition
is a difference in the turn of the phrase. For if it be put in
this way, ' "An animal that walks on two feet" is the definition
of man, is it not ?' or ' "Animal" is the genus of man, is it not ?'
the result is a proposition: but if thus, 'Is "an animal that
walks on two feet" a definition of man or no?' the result is a
problem. Similarly too in other cases. Naturally, then, pro-
blems and propositions are equal in number: for out of every
proposition you will make a problem if you change the turn
of the phrase.
50
ARISTOTLE
Of epoch-making importance is the classification of methods of
proof given in the same connection :
11.04 Having drawn these definitions, we must distinguish
how many species there are of dialectical arguments. There
is on the one hand Induction, on the other Syllogism. Now
what a syllogism is has been said before: induction is a
passage from individuals to universals, e.g. the argument that
supposing the skilled pilot is the most effective, and likewise
the skilled charioteer, then in general the skilled man is the
best at his particular task. Induction is the more convinf ring
and clear: it is more readily learnt by the use of the senses,
and is applicable generally to the mass of men, though
syllogism is more forcible and effective against contradictious
people.
The subject of the Topics are essentially the so-called loci (totcoi).
Aristotle never defined them, and so far no-one has succeeded in
saying briefly and clearly what they are. In any case it is a matter
of certain very general prescriptions for shaping arguments.
An example:
11.05 Now one commonplace rule (totzoc,) is to look and see
if a man has ascribed as an accident what belongs in some other
way. This mistake is most commonly made in regard to the
genera of things, e.g. if one were to say that white happens to
be a colour - for being a colour does not happen by accident to
white, but colour is its genus.
B. PREDICABLES
As an introduction to these loci Aristotle in the first book of the
Topics developed two different doctrines of the structure of state-
ments, both of which obtained considerable historical importance
and still remain of interest: namely the doctrines of the so-called
predicables and of the categories.
11.06 Every proposition and every problem indicates
either a genus or a peculiarity or an accident - for the dif-
ferentia too, applying as it does to a class (or genus), should
be ranked together with the genus. Since, however, of what
is peculiar to anything part signifies its essence, while part
does not, let us divide the 'peculiar' into both the aforesaid
51
THE GREEK VARIETY OF LOGIC
parts, and call that part which indicates the essence a 'defi-
nition', while of the remainder let us adopt the terminology
which is generally current about these things, and speak of
it as a 'property'.
11.07 We must now say what are 'definition', 'property',
'genus', and 'accident'. A 'definition' is a phrase signifying a
thing's essence. It is rendered in the form either of a phrase
in lieu of a term, or of a phrase in lieu of another phrase; for
it is sometimes possible to define the meaning of a phrase as
well.
11.08 A 'property' (iSiov) is a predicate which does not
indicate the essence of a thing, but yet belongs to that thing
alone, and is predicated convertibly of it. Thus it is a property
of man to be capable of learning grammar: for if A be a man,
then he is capable of learning grammar, and if he be capable
of learning grammar, he is a man.
11.09 A 'genus' is what is predicated in the category of
essence of a number of things exhibiting differences in kind.
11.10 An 'accident' is (1) something which though it is
none of the foregoing - i.e. neither a definition nor a property
nor a genus - yet belongs to the thing: (2) something which
may possibly either belong or not belong to any one and the
self-same thing, as (e.g.) the 'sitting posture' may belong or
not belong to some self-same thing.
The logical significance of this division of the 'predicates'
consists in the fact that it is an attempt to analyse propositions, with
reference moreover to the relation between subject and predicate.
This analysis is effected in terms of the matter rather than the form,
yet contains echoes of purely structural considerations, as for instance
in the distinction between genus and specific difference or property,
where the genus is evidently symbolized by a name, properties by a
functor.
As a kind of pendent to the doctrine of the predicables, Aristotle
presents a theory of identity:
11.11 Sameness would be generally regarded as falling,
roughly speaking, into three divisions. We generally apply the
term numerically or specifically or generically - numerically
in cases where there is more than one name but only one
thing, e.g. 'doublet' and 'cloak'; specifically, where there is
more than one thing, but they present no differences in respect
of their species, as one man and another, or one horse and
52
ARISTOTLE
another: for things like this that fall under the same species
are said to be 'specifically the same'. Similarly, too, those
things are called generically the same which fall under the
same genus, such as a horse and a man.
C. CATEGORIES
Another analysis of propositions is contained in the theory of the
categories. This seems to be a systematic development of hints in
Plato. Only in one place (apart from the Categories: 11.12) do we
find an enumeration of ten categories (the only one usually ascribed
to Aristotle):
11.13 Next, then, we must distinguish the classes of
predicates in which the four orders in question (11.06 — 11.10)
are found. These are ten in number: Essence, Quantity, Quali-
ty, Relation, Place, Time, Position, State, Activity, Passivity.
For the accident and genus and property and definition of
anything will always be in one of these categories : for all the
propositions found through these signify either something's
essence or its quality or quantity or some one of the other
types of predicate. It is clear, too, on the face of it that the
man who signifies something's essence signifies sometimes a
substance, sometimes a quality, sometimes some one of the
other types of predicate. For when a man is set before him
and he says that what is set there is 'a man' or 'an animal',
he states its essence and signifies a substance; but when a
white colour is set before him and he says that what is set
there is 'white' or is 'a colour', he states its essence and signi-
fies a quality. Likewise, also, if a magnitude of a cubit be
set before him and he says that what is set there is a magnitude
of a cubit, he will be describing its essence and signifying a
quantity. Likewise also in the other cases.
This text contains an ambiguity: 'essence' (xi kaxi) means first
a particular category - that of substance (oucria) as we see from a
parallel text of the Categories (11.14) - secondly that essence or
intrinsic nature which is found in every category, not only in that of
substance. The thought becomes clear if 'substance' is put in the
list of the ten categories in place of 'essence'.
Here the doctrine of the categories is treated as a division of
sentences and problems for practical purposes. But beyond this
Aristotle regarded it as involving two more important problems.
In the Prior Analytics we read :
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THE GREEK VARIETY OF LOGIC
11.15 The expressions 'this belongs to that' and 'this holds
true of that' must be understood in as many ways as there
are different categories.
That means that the so-called copula of the sentence has as many
meanings as there are categories. That is the first reason why the
theory of the categories is logically so important. The second is
that while this theory constitutes an attempt at classifying objects
according to the ways in which they are predicable, it put in Aristo-
tle's path the problem of the univeral class. He solved it with
brilliant intuition, though, as we now know, with the help of a
faulty proof. The relevant passage occurs in the third book of the
Metaphysics:
11.16 It is not possible that either unity or being should
be a single genus of things; for the differentiae of any genus
must each of them both have being and be one, but it is not
possible for the genus taken apart from its species (any more
than for the species of the genus) to be predicated of its proper
differentiae; so that if unity or being is a genus, no differentia
will either have being or be one.
The line of thought which Aristotle expresses in this very com-
pressed formula is as follows:
1. For all A: if A is a genus, then there is (at least) one B, which
is the specific difference of A;
2. for all A and B: if B is the specific difference of A, then not:
B is A. Suppose now
3. there is an all-inclusive genus V: of this it would be true that
4. for every B: B is V.
As V is a genus, it must have a difference (by 1.); call it D. Of this
D it would be true on the one hand that D is V (by 4.), and on the
other that D is not V (by 2.). Thus a contradiction results, and
at least one of the premisses must be false (cf. 16.33). As Aristotle
holds 1. and 2. to be true, he must therefore reject the supposition
that there is an all-inclusive genus (3.): there is no summum genus.
We have here the basis of the scholastic doctrine of analogy (28.19)
and the first germ of a theory of types (cf. § 47).
The proof is faulty : for 'D is V is not false but meaningless (48.24).
But beyond all doubt the thought confronting us deserves to be
styled a brilliant intuition.
D. SOPHISTIC
The last book of the Topics, known as the Sophistic Befutations,
contains an extensive doctrine of fallacious inferences. Like most
other parts of the Topics this one too belongs to the first form of
Aristotelian logic, not yet formal, but guided by the practical
54
ARISTOTLE
interests of every-day discussion. There is a second doctrine of
fallacious inference, in the Prior Analytics (11.17j, much briefer
than the first but incomparably more formal; all fallacious inferences
are there reduced to breaches of syllogistic laws. However neither
Aristotle himself, nor anyone after him, really succeeded in replacing
the doctrine of the Sophistic Refutations, primitive though it is
from the formal point of view. Knowledge of it is also indispensable
for the understanding of scholastic logic. For all of which reasons
we shall cite a few passages from it here.
11.18 Refutation is reasoning involving the contradictory
of the given conclusion. Now some of them do not really
achieve this, though they seem to do so for a number of
reasons; and of these the most prolific and usual domain is
the argument that turns upon names only. It is impossible in
a discussion to bring in the actual things discussed : we use
their names as symbols instead of them; and therefore we
suppose that what follows in the names, follows in the things
as well, just as people who calculate suppose in regard to
their counters. But the two cases (names and things) are not
alike. For names are finite and so is the sum-total of formulae,
while things are infinite in number. Inevitably, then, the same
formulae, and a single name, have a number of meanings.
Historically, a very important text: in it Aristotle rejects forma-
lism, rightly so for the purposes of ordinary language. For without
preliminary distinction of the various functioning of signs correct
laws cannot be formulated in such a language. The text just cited
underlies the vast growth of medieval doctrine about supposition,
appellation and analogy (§§ 27 and 28). So far as concerns Aristotle
and the other ancient logicians, it appears that they got round the
difficulty mentioned by applying rules by which ordinary language
was turned into an artificial language with a single function for
every verbal form.
11.19 There are two styles of refutation: for some depend
on the language used, while some are independent of language.
Those ways of producing the false appearance of an argument
which depend on language are six in number: they are
ambiguity, amphiboly, combination, division of words,
accent, form of expression.
11.20 Arguments such as the following depend upon ambi-
guity. 'Those learn who know: for it is those who know their
letters who learn the letters dictated to them.' For 'to learn'
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THE GREEK VARIETY OF LOGIC
is ambiguous; it signifies both 'to understand' by the use
of knowledge, and also 'to acquire knowledge'.
11.21 Examples such as the following depend upon amphi-
boly: . . . 'Speaking of the silent is possible': for 'speaking of
the silent' also has a double meaning: it may mean that the
speaker is silent or that the things of which he speaks are so.
11.22 Upon the combination of words there depend
instances such as the following: 'A man can walk while
sitting, and can write while not writing'. For the meaning is
not the same if one divides the words and if one combines
them in saying that 'it is possible to walk-while-sitting . . .'.
The same applies to the latter phrase, too, if one combines the
words 'to write-while-not-writing' : for then it means that he
has the power to write and not to write at once; whereas if
one does not combine them, it means that when he is not
writing he has the power to write.
11.23 Upon division depend the propositions that 5 is 2
and 3, and even and odd, and that the greater is equal: for
it is that amount and more besides.
11.24 Of fallacies, on the other hand, that are independent
of language there are seven kinds :
(1) that which depends upon Accident:
(2) the use of an expression absolutely or not absolutely but
with some qualification of respect, or place, or time, or
relation :
(3) that which depends upon ignorance of what 'refutation'
is:
(4) that which depends upon the consequent:
(5) that which depends upon assuming the original con-
clusion :
(6) stating as cause what is not the cause:
(7) the making of more than one question into one.
And example of (1) is: 'If Coriscus is different from a man he is
different from himself (11.25); of (2): 'Suppose an Indian to be
black all over, but white in respect of his teeth ; then he is both white
and not white' (11.26); (3) consists in proving something other
than what is to be proved (11.27); (5) consists in presupposing what
is to be proved (11.28). (4) alone involves a formal fallacy, namely
concluding from the consequent to the antecedent of a conditional
sentence (11.29).
56
§12. THEORY OF OPPOSITION;
PRINCIPLE OF CONTRADICTION;
PRINCIPLE OF TERTIUM EXCLUSUM
A. THEORY OF OPPOSITION
Aristotle developed two different theories of opposition. The
first, contained in the Topics (12.01) and belonging to the earlier
period of his development, is most clearly summarized in the
pseudo-aristotelian Categories :
12.02 There are four senses in which one thing is said to be
opposed to another: as correlatives, or as contraries, or as
privation and habit (e?i?), or as affirmation and denial. To
give a general outline of these oppositions: the double is
correlative to the half, evil is contrary to good, blindness is a
privation and sight a habit, 'he sits' is an affirmation, 'he does
not sit' a denial.
Two points are worth remark: the division presupposes a material
standpoint, and even in the last case, contradictory opposition,
concerns relationships between terms, not sentences.
It is quite otherwise in the later period. The second doctrine
presupposes the Aristotelian theory of quantification, which is later
than the Topics:
12.03 Some things are universal, others individual. By the
term 'universal' I mean that which is of such a nature as to be
predicated of many subjects, by 'individual' that which is not
thus predicated. Thus 'man' is a universal, 'Callias' an indi-
vidual. . . . If, N then, a man states a positive and a negative
proposition of universal character with regard to a universal,
these two propositions are 'contrary'. By the expression 'a
proposition of universal character with regard to a universal',
such propositions as 'every man is white', 'no man is white'
are meant. When, on the other hand, the positive and negative
propositions, though they have regard to a universal, are yet
not of universal character, they will not be contrary, albeit the
meaning intended is sometimes contrary. As instances of
propositions made with regard to a universal, but not of
universal character, we may take the propositions 'man is
white', 'man is not white'. 'Man 'is a universal, but the pro-
position is not made as of universal character; for the word
'every' does not make the subject a universal, but rather
gives the proposition a universal character. If, however, both
57
THE GREEK VARIETY OF LOGIC
predicate and subject are distributed, the proposition thus
constituted is contrary to truth; no affirmation will, under
such circumstances, be true. The proposition 'every man is
every animal' is an example of this type.
This text contains the following points of doctrine: 1. distinction
between general and singular sentences, according to the kind of
subject; 2. divison of general sentences into universal and particular
according to the extension of the subject; 3. rejection of quantifica-
tion of the predicate. The whole doctrine is now purely formal, and
is explicitly concerned with sentences.
Another division is to be found at the beginning of the Prior
Analytics:
12.04 A premiss then is a sentence affirming or denying one
thing of another. This is either universal or particular or
indefinite. By universal I mean the statement that something
belongs to all or none of something else; by particular that it
belongs to some or not to some or not to all; by indefinite that
it does or does not belong, without any mark to show whether
it is universal or particular.
Here Aristotle enumerates three kinds of sentence: universal,
particular and indefinite. It is striking that no mention is made of
singular sentences. This is due to the fact that every term in the
syllogistic must be available both as subject and predicate, but accord-
ing to Aristotle singular terms cannot be predicated (12.05). In the
particular sentence 'some' means 'at least one, not excluding all'.
Whereas, as Sugihara has recently shewn (12.06), an indefinite
sentence should probably be interpreted in the sense: 'at least one
A is B and at least one A is not B\ However cases in which the
formal properties of particular and indefinite sentences differ are
rare in the syllogistic, so that Aristotle himself often states the
equivalence of these sentences (12.07). Later, even in Alexander of
Aphrodisias (12.08), these cases are dropped altogether.
12.09 Verbally four kinds of opposition are possible, viz.
universal affirmative to universal negative, universal affirma-
tive to particular negative, particular affirmative to universal
negative, and particular affirmative to particular negative:
but really there are only three : for the particular affirmative is
only verbally opposed to the particular negative. Of the
genuine opposites I call those which are universal contraries,
the universal affirmative and the universal negative, e.g. 'all
science is good', 'no science is good'; the others I call contra-
dictories.
58
ARISTOTLE
Here we have the 'logical square' later to become classical, which
can be set out schematically thus:
12.091
to belong to all contrary to belong to none
\
/
% .if*
V
/ * .
/ \
to belong to some only verbal not to belong to all
The logical relationships here intended are shown in the following
passages:
12.10 Of such corresponding positive and negative pro-
positions as refer to universals and have a universal character,
one must be true and the other false.
12.11 It is evident also that the denial corresponding to a
single affirmation is itself single; ... for instance, the affir-
mation 'Socrates is white' has its proper denial in the pro-
position 'Socrates is not white' . . . The denial proper to the
affirmation 'every man is white' is 'not every man is white';
that proper to the affirmation 'some man is white' is 'no man
is white'.
In the later tradition the so-called laws of subalternation also
came to be included in the 'logical square'. They run:
If A belongs to all B, then it belongs to some B (12.12).
If A belongs to no B, then to some B it does not belong (12.13).
B. OBVERSION
In Aristotle's logic negation normally occurs only as a functor
determining a sentence, but there are a number of places in the
Organon where formulae are considered which contain a negation
determining a name. Thus we read in the Hermeneia:
12.14 The proposition 'no man is just' follows from the
proposition 'every man is not-just' and the proposition 'not
every man is not-just', which is the contradictory of 'every
59
THE GREEK VARIETY OF LOGIC
man is not-just', follows from the proposition 'some man is
just'; for if this be true, there must be some just man.
These laws were evidently discovered with great labour and after
experimenting with various false formulae (12.15). Aristotle has
similar ones for sentences with individual subjects in the Hermeneia
(12.16).
In the Prior Analytics Aristotle develops a similar doctrine in
more systematic form and with variables:
12.17 Let A stand for 'to be good', B for 'not to be good',
let C stand for 'to be not-good', and be placed under B, and
let D stand for 'not to be not-good' and be placed under A.
Then either A or B will belong to everything, but they will
never belong to the same thing; and either C or D will belong
to everything, but they will never belong to the same thing.
And B must belong to everything to which C belongs. For if
it is true to say 'it is not-white', it is true also to say 'it is not
white' : for it is impossible that a thing should simultaneously
be white and be not-white, or be a not-white log and be a
white log; consequently if the affirmation does not belong,
the denial must belong. But C does not always belong to B:
for what is not a log at all, cannot be a not-white log either.
On the other hand D belongs to everything to which A belongs.
For either C or D belongs to everything to which A belongs.
But since a thing cannot be simultaneously not-white and
white, D must belong to everything to which A belongs. For
of that which is white it is true to say that it is not not-white.
But A is not true of all D. For of that which is not a log at all
it is not true to say A, viz. that it is a white log. Consequently
D is true, but A is not true, i.e. that it is a white log. It is clear
also that A and C cannot together belong to the same thing,
and that B and D may possibly belong to the same thing.
C. THE PRINCIPLE OF CONTRADICTION
While Aristotle was well acquainted with the principle of identity,
so much discussed later, he only mentions it in passing (12.18). But
to the principle of contradiction he devoted the whole of Book T
of the Metaphysics. This book is evidently a youthful work, and was
perhaps written in a state of excitement, since it contains logical
errors; nevertheless it is concerned with an intuition of fundamental
importance for logic.
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ARISTOTLE
The following are the most important formulations of the principle
of contradiction:
12.19 The same attribute cannot at the same time belong
and not belong to the same subject and in the same respect.
12.20 Let A stand for 'to be good', B for 'not to be good' ....
Then either A or B will belong to everything, but they will
never belong to the same thing.
12.21 It is impossible that contradictories should be at the
same time true of the same thing.
12.22 It is impossible to affirm and deny truly at the same
time.
The first two of these formulae are in the object language, the
last two in a metalanguage, and the author evidently understands
the difference.
In the Topics and Hermeneia Aristotle has a stronger law:
12.23 It is impossible that contrary predicates should belong
at the same time to the same thing.
12.24 Propositions are opposed as contraries when both
the affirmation and the denial are universal ... in a pair of
this sort both propositions cannot be true.
This last statement is quite understandable if one remembers
that when Aristotle was young proofs principally consisted of
refutations. But when Aristotle had developed his syllogistic, in
which refutation has only a subordinate part to play, he not only
found that the principle of contradiction would not do at all as the
first axiom, but also that violence may be done to it in a correct
syllogism.
The first in modern times to advert to this Aristotelian doctrine
was I. Husic in 1906 (12.27). It may seem so astonishing to readers
accustomed to the 'classical' interpretation of Aristotelian logic,
that it is worth while shewing not merely its absolute necessity
but also the context from which our logician's thought clearly
emerges.
12.28 The law that it is impossible to affirm and deny
simultaneously the same predicate of the same subject is not
expressly posited by any demonstration except when the
conclusion also has to be expressed in that form; in which case
the proof lays down as its major premiss that the major
is truly affirmed of the middle but falsely denied. It makes no
difference, however, if we add to the middle, or again to the
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THE GREEK VARIETY OF LOGIC
minor term, the corresponding negative. For grant a minor
term of which it is true to predicate man - even if it be also
true to predicate not-man of it -still grant simply that man is
animal and not not-animal, and the conclusion follows: for
it will still be true to say that Callias - even if it be also true
to say that not-Callias - is animal and not not-animal.
The syllogism here employed has then, omitting quantifiers,
the following form:
12.281 If M is P and not not-P, and £ is M, then S is P
and not not-P.
So the principle of contradiction is no axiom, and does not need
to be presupposed, except in syllogisms of the fore-going kind.
The text quoted is also remarkable in that the middle term in 12.281
is a product (cf. the commentary on 15.151), and that in the minor
term an individual name is substituted (cf. § 13, C, 5) - in each
case contrary to normal syllogistic practice. However the text
comes from the Posterior Analytics and must belong to a relatively
early period.
Aristotle goes still further and states that the principle of con-
tradiction can be completely violated in a conclusive syllogism:
12.29 In the middle (i.e. second: cf. 13.07) figure a syllo-
gism can be made both of contradictories and contraries. Let
'A' stand for 'good', let ' B' and 'C stand for 'science'. If then
one assumes that every science is good, and no science is good,
A belongs to all B and to no C, so that B belongs to no C:
no science is then a science.
This syllogism has the following form :
12.291 If all M is P and no M is P, then no M is M.
D. THE PRINCIPLE OF TERTIUM EXCLUSUM
One formulation of this principle has already been quoted (12.20).
Others are:
12.30 In the case of that which is, or which has taken place,
propositions, whether positive or negative, must be true or
false. Again, in the case of a pair of contradictories, either when
the subject is universal and the propositions are of a universal
character, or when it is individual, as has been said, one of the
two must be true and the other false.
62
ARISTOTLE
12.31 One side of the contradiction must be true. Again, if
it is necessary with regard to everything either to assert or to
deny it, it is impossible that both should be false.
Aristotle's normal practice was to presuppose the correctness of
these theses, and he devoted a notable chapter of the fourth hook of
the Metaphysics (T 8) to the defence of the principle of tertium
exclusion (or tertium non dalur). At least once, however, he called it
into question: in the ninth chapter of the Hermeneia he will not
allow it to be valid for future contingent events. He bases his ar-
gument thus:
12.32 If it is true to say that a thing is white, it must
necessarily be white; if the reverse proposition is true, it
will of necessity not be white .... It may therefore be argued
that it is necessary that affirmations or denials must be
either true or false. Now if this be so, nothing is or takes
place fortuitously, either in the present or in the future, and
there are no real alternatives; everthing takes place of neces-
sity and is fixed. ... It is therefore plain that it is not necessary
that of an affirmation and a denial one should be true and the
other false. For in the case of that which exists potentially,
but not actually, the rule which applies to that which exists
actually does not hold good.
These considerations had no influence on Aristotle's logical
system, as has already been said, but they came to have great
historical importance in the Middle Ages.
The doubt about the validity ot the principle of tertium exclusum
arose from an intuition of the difficult problems which it sets. The
debate is not closed even today.
§13. ASSERTORIC SYLLOGISTIC
We give here a page of the Prior Analytics in as literal a trans-
lation as possible, and comment on it afterwards. It contains the
essentials of what later came to be called Aristotle's assertoric
syllogistic. It is a leading text, in which no less than three great
discoveries are applied for the first time in history: variables, purely
formal treatment, and an axiomatic system. It constitutes the
beginning of formal logic. Short as it is, it formed the basis of logical
speculation for more than two thousand years - and yet has been
only too often much misunderstood. It deserves to be read atten-
tively.
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THE GREEK VARIETY OF LOGIC
A. TEXT
Aristotle begins by stating the laws of conversion of sentences.
These are cited below, 14.10 ff., among the bases of the systematic
development. He goes on :
13.01 When then three terms are so related one to another
that the last is in the middle (as in a) whole and the middle is or is
not in the first as in a whole, then there must be a perfect
syllogism of the two extremes.
13.02 Since if A (is predicated) of all B, and B of all C, A
must be predicated of all C.
13.03 Similarly too if A (is predicated) of no B, and B of
all A, it is necessary that A will belong to no C.
13.04 But if the first follows on all the middle whereas the
middle belongs to none of the last, there is no syllogism of the
extremes; for nothing necessary results from these; for the
first may belong to all and to none of the last; so that neither
a particular nor a universal is necessary; and since there is
nothing necessary these produce no syllogism. Terms for
belonging to all: animal, man, horse; for belonging to none:
animal, man, stone.
13.05 But if one of the terms is related wholly, one partially,
to the remaining one; when the wholly related one is posited
either affirmatively or negatively to the major extreme, and
the partially related one affirmatively to the minor extreme,
there must be a perfect syllogism . . . for let A belong to all B
and B to some C, then if being predicated of all is what has
been said, A must belong to some C.
13.06 And if A belongs to no B and B to some C, to some
C A must not belong . . . and similarly if the BC (premiss) is
indefinite and affirmative.
13.07 But when the some belongs to all of one, to none of
the other; such a figure I call the second.
13.08 For let M be predicated of no N and of all X ; since
then the negative converts, N will belong to no M ; but M
was assumed (to belong) to all X ; so that N (will belong) to
no X ; for this has been shewn above.
13.09 Again, if M (belongs) to all iV and to no X, X too
will belong to no N; for if M (belongs) to no X, X too (belongs)
to no M ; but M belonged to all N ; therefore X will belong to
no iV, for the first figure has arisen again ; but since the nega-
tive converts, N too will belong to no X, so that it will be the
64
ARISTOTLE
same syllogism. It is possible to shew this by bringing to
impossibility.
13.10 If M belongs to no TV and to some X, then to some
X TV must not belong. For since the negative converts, TV
will belong to no M, but M has been supposed to belong to
some X; so that to some X TV will not belong; for a syllogism
arises in the first figure.
13.11 Again, if M belongs to all TV and to some X not. to
some TV X must not belong. For if it belongs to all, and M is
predicated of all TV, M must belong to all X ; but to some it has
been supposed not to belong.
13.12 If to the same, one belongs to all, the other to none, or
both to all or to none, such a figure I call third.
13.13 When both P and R belong to all S, of necessity P
will belong to some B; for since the affirmative converts, S
will belong to some P, so that when P (belongs) to all S, and
S to some R, P must belong to some R; for a syllogism
arises in the first figure. One can make the proof also by (bring-
ing to) the impossible and by setting out (terms); for if both
belong to all S, if some of the (things which are) S be taken, say TV,
to this both P and R will belong, so that to some R P will belong.
13.14 And if R belongs to all S, and P to none, there will
be a syllogism that to some R P necessarily will not belong.
For there is the same manner of proof, with the RS premiss
converted. It could also be shewn by the impossible as in the
previous cases.
13.15 If R (belongs) to all S and P to some, P must belong
to some R. For since the affirmative converts, 8 will belong
to some P, so that when R (belongs) to all S, and S to some P,
R too will belong to some P; so that P (will belong) to some R.
13.16 Again, if R belongs to some S and P to all, P must
belong to some R; for there is the same manner of proof.
One can also prove it by the impossible and by setting out,
as in the previous cases.
13.17 If R belongs to all S, and to some (S) P does not.
then to some R P must not belong. For if to all, and R (belongs)
to all S, P will also belong to all S; but it did not belong. It is
also proved with reduction (to the impossible) if some of
what is S be taken to which P does not belong.
13.18 If P belongs to no S, and R to some S, to some R P
will not belong; for again there will be the first figure when the
RS premiss is converted.
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B. INTERPRETATION
This passage is composed in such compressed language that most
readers find it very hard to understand. Indeed the very style is of
the greatest significance for the history of logic ; for here we have the
manner of thought and writing of all genuine formal logicians, be
they Stoics or Scholastics, be their name Leibniz or Frege. Hence
we have given a literal version, but shall now interpret it with the
aid of paraphrase and commentary:
on 13.01 : Aristotle defines the first figure. This may serve as an
example: Gainful art is contained in art in general as in a whole;
the art of pursuit (e.g. hunting) is contained in gainful art as in
a whole; therefore the art of pursuit is contained in art in general
as in a whole. The example is taken from Plato's division (8.05),
from which the Aristotelian syllogism seems to have developed.
We shall explain what a perfect syllogism is in § 14.
on 13.02: This mood later (with Peter of Spain) came to be called
'Barbara'. Hereafter we refer to all moods by the mnemonic names
originating with Peter of Spain (cf. 32.04 ff.).
We obtain an example by substitution:
If animal belongs to all man
and man belongs to all Greek,
then animal belongs to all Greek.
on 13.03: Celarent: If stone belongs to no man
and man belongs to all Greek,
then stone belongs to no Greek.
on 13.04: Names are here given for two substitutions by which
it can be shewn that a further mood is invalid. Probably the follow-
ing are intended :
Mood Substitution
If A to all B 1. If animal belongs to all man
and B to no C, and man belongs to no horse,
then A to no C, then animal belongs to no horse.
2. If animal belongs to all man
and man belongs to no stone,
then animal belongs to no stone.
In each case the premisses are true, but the conclusion is once
true, once false. Therefore the mood is invalid.
on 13.05: Darii: If Greek belongs to all Athenian
and Athenian belongs to some logician,
then Greek belongs to some logician.
It is to be noticed that here and subsequently 'some' must have
the sense of 'at least one'.
on 13.06: Ferio: If Egyptian belongs to no Greek
and Greek belongs to some logician,
then to some logician Egyptian does not belong.
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ARISTOTLE
on 13.07: Aristotle defines the second figure, in which the middle
term is predicate in both premisses. He considers three cases: 1. one
premiss is universal and affirmative, the other universal and negative,
2. both premisses are universal and affirmative, '.>. both premisses
are negative. Only in the first case there are valid syllogisms,
on 13.08: Cesare: If man belongs to no stone
and man belongs to all Greek,
then stone belongs to no Greek.
This is reduced to Celarent (13.03) by conversion of the major
(first) premiss: If stone belongs to no man
and man belongs to all Greek,
then stone belongs to no Greek,
on 13.09: Camestres: If (1) animal belongs to all man
and (2) animal belongs to no stone,
then (3) man belongs to no stone.
The proof proceeds by reduction to Celarent (13.03). First the minor
premiss (2) is converted :
(4) stone belongs to no animal;
then comes the other premiss :
(1) animal belongs to all man.
(4) and (1) are the premisses of Celarent, from which follows the
conclusion,
(5) stone belongs to no man. This is converted, and so the desired
conclusion is obtained.
It is important to notice that this conclusion is first stated by
Aristotle at the end of the process of proof.
on 13.10: Festino: The proof of this mood is by reduction to Ferio
(13.06), the major premiss being converted as in Cesare (13.08).
on 13.11: Baroco: If (1) Greek belongs to all Athenian
and (2) to some logician Greek does not belong,
then (3) to some logician Athenian does not
belong.
The proof proceeds by first hypothesizing the conclusion as false,
i.e. its contradictory opposite is supposed :
(4) Athenian belongs to all logician.
Now comes the first (major) premiss:
(5) Greek belongs to all Athenian.
From these one obtains a syllogism in Barbara (13.02):
If (5) Greek belongs to all Athenian
and (4) Athenian belongs to all logician,
then (6) Greek belongs to all logician.
But the conclusion (6) of this syllogism is contradictorily opposed
to the minor premiss of Baroco, (2), and as this is accepted, the former
must be rejected. So one of the premisses (4) and (5) must be rejected :
as (5) is accepted, (4) must be rejected; and so we obtain the con-
tradictory opposite of (4), namely (3).
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THE GREEK VARIETY OF LOGIC
on 13.12: Aristotle defines the third figure, in which the middle
term is subject in both premisses. He considers the same three
cases as in 13.07.
on 13.13: Darapti: If Greek belongs to all Athenian
and man belongs to all Athenian,
then Greek belongs to some man.
This syllogism is first reduced to Darii (13.05) by conversion of the
minor premiss - just as Cesare (13.08) is reduced to Celarent (13.03).
But Aristotle then develops two further methods of proof: a process
- as with Baroco (13.11) - 'through the impossible', and the 'setting
out of terms'. This last consists in singling out a part, perhaps an
individual (but this is debated in the literature), from the Athenians,
say Socrates. It results that as Greek as well as man belongs to all
Athenian, Socrates must be Greek as well as man. Therefore this is
a Greek who is man. Accordingly Greek belongs to (at least) one
man.
on 3.14: Felapton: If Egyptian belongs to no Athenian
and man belongs to all Athenian,
then to some man Egyptian does not belong.
The proof proceeds by conversion of the minor (second) premiss,
resulting in Ferio (13.06).
on 3.15: Disamis: If Athenian belongs to all Greek
and logician belongs to some Greek,
then logician belongs to some Athenian.
The first thing to be noticed is that Aristotle here writes the minor
premiss first, as also in 13.16 and 13.17. The proof is by reduction
to Darii (13.05), just as Camestres (13.09) was reduced to Celarent
(13.03).
on 13.16: Datisi: If logician belongs to some Greek
and Athenian belongs to all Greek,
then Athenian belongs to some logician.
The proof proceeds by reduction to Darii (13.05), the minor premiss
(here the first!) being converted.
on 13.17: Bocardo: If Greek belongs to all Athenian
and to some Athenian logician does not
belong,
then to some Greek logician does not belong.
The premisses are again in reversed order, the minor coming first.
The proof proceeds by reduction to the impossible, with use of
Barbara (13.02) as in the case of Baroco (13.11). A proof by setting
out of terms is also recommended, but not carried through,
on 13.18: Ferio: If Egyptian belongs to no Greek
and logician belongs to some Greek,
then to some logician Egyptian does not belong.
Aristotle reduces this syllogism to Ferio (13.06), by conversion of
the minor (second) premiss.
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ARISTOTLE
This paraphrase with comments is, be it noted, a concession to the
modern reader. For in his Analytics Aristotle never argued by means
of substitutions, as we have been doing, except in proofs of invalidity.
However, in view of the contemporary state of logical awareness, it
seems necessary to elucidate the text in this more elementary way.
C STRUCTURE OF THE SYLLOGISM
If we consider the passages on which we have just commented,
the first thing we notice is that the definition which Aristotle gives of
the syllogism (10.05), does indeed contain it, but is much too wide:
the analytic syllogism as we call the class of formulae considered in
chapters 4-6 of the first book of the Prior Analytics, can be described
as follows:
1. It is a conditional sentence, the antecedent of which is a con-
junction of two premisses. Its general form is : ' If p and q, then r', in
which propositional forms are to be substituted for 'p', 'g' and V. So
the Aristotelian syllogism has not the later form: 'p; q; therefore r',
which is a rule. The Aristotelian syllogism is not a rule but a proposition.
2. The three propositional forms whose inter-connection produces a
syllogism, are always of one of these four kinds: 'A belongs to all B\
A belongs to no B\ lA belongs to (at least) some B\ ' (at least) to
some B A does not belong'. Instead of this last formula, there some-
times occurs the (equivalent) one: 'A belongs not to all B\ The
word 'necessary' or 'must' is often used : evidently that only means
here that the conclusion in question follows logically from the
premisses.
3. Where we have been speaking not of propositions but of
forms, seeing that Aristotelian syllogisms always contain letters
('A', 'B', T' etc.) in place of words, which are evidently to be
interpreted as variables, Aristotle himself gives examples of how
substitutions can be made in them. That is indeed the only kind
of substitution for variables known to him: he has, for example,
no thought of substituting variables for variables. Nevertheless
this is an immense discovery : the use of letters instead of constant
words gave birth to formal logic.
4. In every syllogism we find six such letters, called 'terms'
(opot., 'boundaries'), equiform in pairs. Aristotle uses the following
terminology: the term which is predicate in the conclusion and the
term equiform to it in one of the premisses, are called 'major',
evidently because in the first figure - but there only - this has the
greatest extension. The term which is subject in the conclusion, and
the term equiform to it in one of the premisses, he calls a 'minor' or
'last' term (eXocttov, Ict^octov) for the same reason. Finally the two
equiform terms that occur only in the premisses are called 'middle'
terms. By contrast the two others are called 'extremes' (axpa).
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THE GREEK VARIETY OF LOGIC
Admittedly the terms are not so defined in the text of Aristotle : he
gives complicated definitions based on the meaning of the terms ; but
his customary syllogistic practice keeps to the foregoing definitions.
- Sometimes the premiss containing the major term is called 'the
major', the other 'the minor'.
5. The letters (variables) can, in the system, only be substituted by
universal terms; they are term-variables for universal terms. But it
would not be right to call them class-variables, for that would be to
ascribe to Aristotle a distinction between intension and extension
which is out of place. One may ask why the founder of logic, whose
philosophical development proceeded steadily away from Platonism
towards a recognition of the importance of the individual, completely
omitted singular terms in what (by contrast to the Hermeneia) is
his most mature work. The reason probably lies in his assumption
that such terms are not suitable as predicates (13.19), whereas
syllogistic technique requires every extreme term to occur at least
once as predicate.
6. It is usually said that a further limitation is required, namely
that void terms must not be substituted for the variables. But this is
only true in the context of certain interpretations of the syllogistic;
on other interpretations this limitation is not required.
To sum up : we have in the syllogistic a formal system of term-logic,
with variables, limited to universal terms, and consisting of propo-
sitions, not rules.
This system is also axiomatized. Hence we have here together
three of the greatest discoveries of our science: formal treatment,
variables, and axiomatization. What makes this last achievement
the more remarkable is the fact that the system almost achieves
completeness (there is lacking only an exact elaboration of the
moods of the fourth figure). This is something rare for the first
formulation of a quite original logical discovery.
D. THE FIGURES AND FURTHER SYLLOGISMS
The syllogisms are divided into three classes (ax^aTa), 'figures' as
this was later translated, according to the position of the middle term.
According to Aristotle there are only three such figures :
13.20 So we must take something midway between the
two, which will connect the predications, if we are to have a
syllogism relating this to that. If then we must take something
common in relation to both, and this is possible in three ways
(either by predicating A of C, and C of B, or C of both, or
both of C), and these are the figures of which we have spoken,
it is clear that every syllogism must be made in one or other of
these figures.
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ARISTOTLE
But evidently this is so far from being the case that Aristotle
himself was well aware of a fourth figure. He treats its syllogisms
as arguments obtainable from those already gained in the first
figure:
13.21 It is evident also that in all the figures, whenever a
proper syllogism does not result, if both the terms are affir-
mative or negative nothing necessary follows at all, but if one
is affirmative, the other negative, and if the negative is stated
universally, a syllogism always results relating the minor to
the major term, e.g. if A belongs to all or some B, and B
belongs to no C: for if the premisses are converted it is
necessary that C does not belong to some A. Similarly also in
the other figures.
The case under consideration is this: the premisses are (1) lA
belongs to all B\ (2) 'B belongs to no C\ They are both converted and
their order is reversed (an operation that is irrelevant for Aristotle),
so that we get: lC belongs to no B' and 'B belongs to some A\ But
those are the premisses of the fourth mood of the first figure (Ferio,
13.06), which has as conclusion 'to some A C does not belong'. Now
if 'major term' and 'minor term' are defined as has been done above
(§ 13, C 4) in accordance with the practice of Albalag and the moderns,
then evidently C is the major term, A the minor, and so (2) is the
major premiss, (1) the minor premiss. From that it follows that the
middle term is predicate in the major premiss, subject in the minor,
just the reverse of the situation in the first figure. We have here
therefore a fourth figure. That Aristotle refuses to recognize any
such, is due to his not giving a theoretical definition of the terms
according to their place in the conclusion, but according to their
extension, and so not a formal definition but one dependent on their
meaning. - The syllogism just investigated later came to be called
Fresison.
Aristotle explicitly stated two syllogism of this figure, Fresison
already cited, and in the same passage (13.31) Fesapo; he hints only
at three more (13.32): Dimaris, Bamalip and Camenes.
The same text (13.22) would permit us to gain still other syllo-
gisms from two of the second figure Cesare: 13.08 and Cameslres:
13.09) and from three of the third {Darapti: 13.13, Disamis: 13.15
and Dalisi: 13.16). It is worth remarking that these hints do not
occur in the text of the proper presentation of the syllogistic;
apparently Aristotle only made these discoveries when his system
was already in being.
Consequently the following passage seems to be a later addition.
It is at the origin of what later came to be called the 'subalternate
syllogisms', Barbari, Celaront, Cesaro, Camestrop and Calemop.
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THE GREEK VARIETY OF LOGIC
13.23 It is possible to give another reason concerning
those (syllogisms) which are universal. For all the things that
are subordinate to the middle term or to the conclusion may
be proved by the same syllogism, if the former are placed in
the middle, the latter in the conclusion; e.g. if the conclusion
AB is proved through C, whatever is subordinate to B or C
must accept the predicate A : for if D is included in B as in a
whole, and B is included in A, then D will be included in A.
Again if E is included in C as in a whole, and C is included in
A, then E will be included in A. Similarly if the syllogism is
negative. In the second figure it will be possible to infer only
that which is subordinate to the conclusion, e.g. if A belongs
to no B and to all C ; we conclude that B belongs to no C. If
then D is subordinate to C, clearly B does not belong to
it. But that B does not belong to what is subordinate to A, is
not clear by means of the syllogism.
These syllogisms, however, are not developed.
If we want to summarize the content of the texts we have adduced,
we see that Aristotle in fact expressly formulated the conditions
required for a system of twenty-four syllogistic moods, six in each
figure. Of these twenty-four he only developed nineteen himself,
only fourteen of them thoroughly. The remaining ten fall into three
classes: (1) exactly formulated (Fesapo, Fresison: 13.21): (2) not
formulated, but clearly indicated (Dimaris, Bamalip, Camenes:
13.22); (3) only indirectly indicated: the five 'subalternate' moods
(13.23).
That explains how historically sometimes fourteen, sometimes
nineteen and again at other times twenty-four moods are spoken of.
The last figure is obviously the only correct one. For evidently no
systematic principle can be derived from the fact that the author of
the syllogistic did not precisely develop certain moods.
§ 14. AXIOMATIZATION OF THE SYLLOGISTIC
FURTHER LAWS
A. AXIOMATIC THEORY OF THE SYSTEM
Aristotle axiomatized the syllogistic, and in more than one way. In
this connection we shall first cite the most important passages in
which he presents his theory of the system as axiomatized, and then
give the axiomatization itself. For this theory is the first of its kind
known to us, and notwithstanding its weaknesses, must be consi-
dered as yet another quite original contribution made to logic by
72
ARISTOTLE
Aristotle. The point is a methodological one of course, not a matter of
formal logic, and that Aristotle was himself aware of:
14.01 We now state by what means, when, and how every
syllogism is produced; subsequently we must speak of demon-
stration. Syllogism should be discussed before demonstration,
because syllogism is the more general: demonstration is a
sort of syllogism, but not every syllogism is a demonstration.
Aristotle's doctrine of demonstration is precisely his methodology.
But as the methodology of deduction is closely connected with formal
logic, we must go into at least a few details.
14.02 We suppose ourselves to possess unqualified scientific
knowledge of a thing, as opposed to knowing it in the accidental
way in which the sophist knows, when we think that we know
the cause on which the fact depends, as the cause of that fact
and of no other, and, further, that the fact could not be other
than it is . . . There may be another manner of knowing as
well - that will be discussed later. What I now assert is that
at all events we do know by demonstration. By demonstration
I mean a syllogism productive of scientific knowledge, a
syllogism, that is, the grasp of which is eo ipso such knowledge.
Assuming then that my thesis as to the nature of scientific
knowing is correct, the premisses of demonstrated knowledge
must be true, primary, immediate, better known than and
prior to the conclusion, which is further related to them as
effect to cause. Unless these conditions are satisfied, the basic
truths will not be 'appropriate' to the conclusion. Syllogism
there may indeed be without these conditions, but such
syllogism, not being productive of scientific knowledge, will
not be demonstration.
14.03 There are three elements in demonstration : (1)
what is proved, the conclusion - an attribute inhering essen-
tially in a genus; (2) the axioms, i.e. the starting points of
proof; (3) the subject-genus whose attributes, i.e. essential
properties, are revealed by the demonstration.
It emerges clearly from this text that for Aristotle a demonstration
(1) is a syllogism, (2) with specially constructed premisses, and (3)
with a conclusion in which a property (11.081 is predicated of a genus.
That, however, can only be achieved by means of a syllogism in the
first figure :
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THE GREEK VARIETY OF LOGIC
14.04 Of all figures the most scientific is the first. Thus, it
is the vehicle of the demonstrations of all the mathematical
sciences, such as arithmetic, geometry, and optics, and practi-
cally of all sciences that investigate causes ... a second proof
that this figure is the most scientific; for grasp of a reasoned
conclusion is the primary condition of knowledge. Thirdly,
the first is the only figure which enables us to pursue know-
ledge of the essence of a thing. . . . Finally, the first figure has
no need of the others, while it is by means of the first that the
other two figures are developed, and have their intervals
close-packed until immediate premisses are reached. Clearly,
therefore, the first figure is the primary condition of knowledge.
This doctrine is only of historical importance, though that is
considerable: on the other hand the essential of Aristotle's views on
the structure of an axiomatic system has remained a part of every
methodology of deduction right to our own day:
14.05 Our own doctrine is that not all knowledge is demon-
strative: on the contrary, knowledge of the immediate
premisses is independent of demonstration. (The necessity of
this is obvious; for since we must know the prior premisses
from which the demonstration is drawn, and since the regress
must end in immediate truths, those truths must be indemon-
strable.) . . Now demonstration must be based on premisses
prior to and better known than the conclusion ; and the same
things cannot simultaneously be both prior and posterior to
one another: so circular demonstration is clearly not possible
in the unqualified sense of 'demonstration', but only possible
if 'demonstration' be extended to include that other method
of argument which rests on a distinction between truths
prior to us and truths without qualification prior, i.e. the
method by which induction produces knowledge. . . . The
advocates of circular demonstration are not only faced with
the difficulty we have just stated: in addition their theory
reduces to the mere statement that if a thing exists, then it
does exist - an easy way of proving anything. That this is
so can be clearly shown by taking three terms, for to constitute
the circle it makes no difference whether many terms or few
or even only two are taken. Thus by direct proof, if A is, B
must be ; if B is, C must be ; therefore if A is, C must be. Since
then - by the circular proof - if A is, B must be, and if B is, A
must be, A may be substituted for C above. Then 'if B is, A
74
ARISTOTLE
must be' = 'if B is, C must be', which above gave the con-
clusion 'if A is, C must be', but C and A have been identified.
This is, be it said at once, far the clearest passage about our
problem, which evidently faced Aristotle with enormous difficulties.
Two elements are to be distinguished : on the one hand it is a matter
of epislemological doctrine, according to which all scientific knowledge
must finally be reduced to evident and necessary premisses; on the
other hand is a logical theory of deduction, which states that one
cannot demonstrate all sentences in a system, but must leave off
somewhere; for neither a processus in infinitum nor a circular
demonstration is possible. In other words: there mu-t be axioms in
every system.
B. SYSTEMS OF SYLLOGISTIC
This doctrine was now applied by Aristotle to formal logic itself,
i.e. to the syllogistic ; yes, the syllogistic is the first known axiomatic
system, or more precisely the first class of such systems: for Ari-
stotle axiomatized it in several ways. One can distinguish in his
work the following systems: 1) with the four syllogisms of the first
figure (together with other laws) as axioms, 2) with the first two
syllogisms of the same figure, 3) with syllogisms of any figure as
axioms, in which among other features the syllogisms of the first
figure are reduced to those of the second and third. These three
systems are presented in an object-language; there is further to be
found in Aristotle a sketch for the axiomatization of the syllogistic
in a metalanguage.
We take first the second system, the first having been fully
presented above in § 13.
14.06 It is possible also to reduce all syllogisms to the
universal syllogisms in the first figure. Those in the second
figure are clearly made perfect by these, though not all in the
same way; the universal syllogisms are made perfect by con-
verting the negative premiss, each of the particular syllogisms
by reductio ad impossibile. In the first figure particular
syllogisms are indeed made perfect by themselves, but it is
possible also to prove them by means of the second figure,
reducing them ad impossibile, e.g. if A belongs to all B, and
B to some C, it follows that A belongs to some C. For if it
belonged to no 67, and belongs to all B, then B will belong
to no C: this we know by means of the second figure.
It is here shewn that Darii (13.05) can be reduced to Cameslres
(13.09) ; the proof of Ferio (13.06) is similarly effected, and it is then
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THE GREEK VARIETY OF LOGIC
shewn that the syllogisms of the third figure can also be easily
deduced.
In these operations the syllogisms of the first figure always play
the part of axioms, for the reason that they are to be 'perfect'
(TeAeLOi) syllogisms (14.08). This expression is explained thus:
14.09 I call that a perfect syllogism which needs nothing
other than what has been stated to make plain what neces-
sarily follows; a syllogism is imperfect, if it needs either one
or more propositions, which are indeed the necessary conse-
quences of the terms set down, but have not been expressly
stated as premisses.
But this can only mean that perfect syllogisms are intuitively
evident.
C. DIRECT PROOF
To be able to deduce his syllogisms Aristotle makes use of three
procedures, and in each of another class of formulae, not named as
axioms and in part tacitly presented. The procedures are direct
proof (SzixTitic, avaysLv), reduction to the impossible (zlc, to aSuvarov
avayetv) and ecthesis (setting out of terms, exOsctic;).
In direct proof the laws of conversion of sentences are explicitly
presupposed ; they are three :
14.10 If A belongs to no B, neither will B belong to any A.
For if to some, say to C, it will not be true that A belongs to
no B; for C is one of the things (which are) B.
14.11 If A belongs to all B, B also will belong to some A;
for if to none, then neither will A belong to any B; but by
hypothesis it belonged to all.
14.12 If A belongs to some B, B also must belong to some
A; for if to none, then neither will A belong to any B.
Aristotle prefaces the syllogistic proper with these laws and their
justification, clearly conscious that he needs them for the 'direct
procedure' .They are the laws of conversion of affirmative (universal
and particular), and universal negative propositions. (The conversion
of particular negatives is expressly recognized as invalid: 14.13.)
It is noteworthy that Aristotle tries to axiomatize these laws too :
the first is proved by ecthesis and serves as axiom for the two
others.
Besides these explicit presuppositions of the syllogistic, some
rules of inference are also used, without Aristotle having consciously
reflected on them. They are these :
14.141 Should 'If p and q, then r' and 'If s, then p' be
valid, then also 'If s and q, then r' is valid.
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ARISTOTLE
14.142 Should 'If p and q, then r' and 'If s, then qf be
valid, then also 'If p and s, then r' is valid.
14.151 Should 'If p and q, then r' be valid, then also 'If q
and p, then r' is valid.
14.161 Should 'If p, then </' and 'If </, then r' be valid, then
also 'If p, then r' is valid.
Some of these rules also used - without being explicitly appealed
to - for constructing formulae that later came to be called 'polysyllo-
gisms, or 'soriteses' :
14.17 It is clear too that every demonstration will proceed
through three terms and no more, unless the same con-
clusion is established by different pairs of propositions; . . .
Or again when each of the propositions A and B is obtained
by syllogistic inference, e.g. A by means of D and E, and
again B by means of F and G. . . . But thus also the syllogisms
are many; for the conclusions are many, e.g. A and B and C.
Also to be noted in this text is Aristotle's evident use of proposi-
tional variables.
D. INDIRECT PROOF
Aristotle has two different procedures for reduction to the impos-
sible, the first being invalid and clearly earlier. In both, the laws of
opposition are presupposed. By contrast to the laws of conversion
these are neither systematically introduced nor axiomatized; they
occur as the occassion of the deduction requires. The reason for
their not being systematized or axiomatized may be that the essential
points about them have been stated already in the Hermeneia. The
main features have been summarized above (12.10, 12,11).
The procedures are as follows:
First procedure
It is used to reduce Baroco (13.11) and Bocardo (13.17), and takes
the course outlined above (in the commentary on 13.11). As Luka-
siewicz (14.18) has shown, it is not conclusive. This can be made
evident by the following substitution: we put 'bird' for 'M', 'beast'
for W and 'owl' for lX' in Baroco (13.11). We obtain:
If (1) bird belongs to all beast
and (2) to some owl bird does not belong,
then (3) to some owl beast does not belong.
The syllogism is correct, being a substitution in Baroco; but all its
three component sentences are manifestly false. Now if we apply the
procedure described above (commentary on 13.11), we must form
the contradictory opposite to (3) :
(4) Beast belongs to all owl.
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THE GREEK VARIETY OF LOGIC
This produces with (1) a syllogism in Barbara (13.02), having as
conclusion
(5) Bird belongs to all owl, which so far from being false, is
evidently true. Hence the procedure fails to give the required
conclusion and must be deemed incorrect.
It would certainly be correct if Aristotle had not expressed the
syllogism as a conditional sentence (in which the antecedent does not
need to be asserted), but in the scholastic manner as a rule (31.11)
in which one starts from asserted premisses.
Second procedure
We do not know whether Aristotle saw the incorrectness of the
first procedure; in any case in book B of the Prior Analytics he
several times uses another which is logically correct.
It is to be found in the place where he treats of the so-called
'conversion' (avTLaTpocp-y)) of syllogisms, a matter of replacing one
premiss by the (contradictory) opposite of the conclusion.
14.19 Suppose that A belongs to no B, and to some C:
the conclusion is BC. ... If the conclusion is converted into
its contradictory, both premisses can be refuted. For if B
belongs to all C, and A to no B, A will belong to no C: but
it was assumed to belong to some C.
The following scheme reproduces the thought:
Original syllogism Converted syllogism
(Festino) (Celarent)
If A belongs to no B If A belongs to no B
and A belongs to some C, and B belongs to all C,
then to some C B does not then A belongs to no C
belong.
The rule presupposed, of which Aristotle was conscious (cf.
16.33) - he often used it, - is this:
14.201 Should 'If p and q, then r' be valid then also 'If not-r
and q, then not-p' is valid.
A similar rule, also often used, is:
14.202 Should 'If p and q, then r' be valid, then also 'If
p and not-r, then not-qr' is valid.
By the use of these rules with the laws of opposition and some of
the rules given above (14.151, 14.161), any syllogism can in fact be
reduced to another.
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ARISTOTLE
Applications
By varying this second procedure Aristotle was able to construct
a third axiomatization - or rather a class of further axiomatiza-
tions - of his syllogistic: Syllogisms of either the first, second or
third figure are taken, and the others proved from them by reduc-
tion to the impossible. The result is summarized thus:
14.21 It is clear that in the first figure the syllogisms are
formed through the middle and the last figures ... in the
second through the first and the last figures ... in the third
through the first and the middle figures.
We shall not go into the practical details (14.22), but only note
that Aristotle replaces premisses not only by their contradictory,
but also by their contrary opposites, and that he investigates all
syllogisms.
The results of replacing premisses by the contradictory opposite
of the conclusion can be clearly presented in the following way:
From a syllogism of figure 1 2 3
there results by substitution
of the negation of the conclusion
a syllogism of figure:
substitution for the major premiss 3 3 1
substitution for the minor premiss 2 12
E. DICTUM DE OMNI ET NULLO
A word must now be said about the 'dictum de omni el nullo' that
later became so famous. It concerns the following sentence:
14.23 That one thing should be in the whole of another and
should be predicated of all of another is the same. We say
that there is predication of all when it is impossible to take
anything of which the other will not be predicated; and
similarly predication of none.
It is not clear whether Aristotle really intended here to establish
an axiom for his system, as has often been supposed. One is rather
led to suppose that he is simply describing the first and second
moods of the first figure (13.02, 13.03). However, the dictum can
be understood as an axiom if it is considered as a summary of the
first four moods of the first figure, which is not in itself impossible.
In this connection we quote a historically and systematically
more important passage in which Aristotle deals with a problem of
the theory of the three figures (13.20). In it he makes an essential
advance in analyzing propositions and gives expression to thoughts
that are not without significance for the theory of quantification.
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THE GREEK VARIETY OF LOGIC
14.24 It is not the same, either in fact or in speech, that A
belongs to all of that to which B belongs, and that A belongs
to all of that to all of which B belongs : for nothing prevents B
from belonging to C, though not to all C: e.g. let B stand
for 'beautiful' and C for 'white'. If beauty belongs to something
white, it is true to say that beauty belongs to that which is
white: but not perhaps to everything that is white.
Here an analysis of the sentence A belongs to all B 'is presented,
which could be interpreted in this way: 'For all x: if B belongs to x,
then A belongs to x; it would then be a matter of the modern formal
implication. That Aristotle thought of such an analysis - at least
during his later period - seems guaranteed by the fact that he
explicitly applied it to modal logic (cf. 15.13). The Scholastics, as we
shall see, treated these thoughts as an elucidation of the dictum.
F. BEGINNINGS OF A METALOGICAL SYSTEM
Aristotle also described his syllogisms metalogically in such a way
that a new, metalogical system could easily be established:
14.25 In every syllogism one of the premisses must be
affirmative, an universality must be present.
14.26 It is clear that every demonstration will proceed
through three terms and no more, unless the same conclusion
is established by different pairs of propositions.
14.27 This being evident, it is clear that a syllogistic
conclusion follows from two premisses and not from more
than two.
14.28 And it is clear also that in every syllogism either
both or one of the premisses must be like the conclusion. I
mean not only in being affirmative or negative, but also in
being necessary, assertoric or contingent.
Aristotle does not carry out this application to modal logic;
possibly this is an interpolation by another hand.
In developing the several figures Aristotle established similar
rules for each. 13.05 contains an example. Taken all together these
rules form an almost complete metalogical description of the syllo-
gistic, which one would like to develop.
G. THE INVENTIO MEDII
Here we want to allude briefly to a doctrine of the Prior Analytics
which is not essentially a matter of formal logic but rather of
methodology, and that is the discussions about what was later
80
ARISTOTLE
called the inventio medii. In connection with axiomatization one
can ask two different b;isic questions: (1) What follows given
premisses? (2) From what premisses can ;i given sentence (con-
clusion) be deduced? Aristotle primarily considered the first ques-
tion, but in the following text and its continuation he poses also
the second, and tries to show the premisses of a syllogism must be
constructed in order to yield a given conclusion. At the same time
he gives practical advice on the forming of syllogisms:
14.29 The manner in which every syllogism is produced,
the number of the terms and premisses through which il
proceeds, the relation of the premisses to one another, the
character of the problem proved in each figure, and the num-
ber of the figures appropriate to each problem, all these
matters are clear from what has been said. We must now
state how we may ourselves always have a supply of syllogisms
in reference to the problem proposed and by what road we may
reach the principles relative to the problem: for perhaps we
ought not only to investigate the construction of syllogisms,
but also to have the power of making them.
We do not need to pursue the details of this theory here. It only
interests us as the starting-point of the scholastic pons asinorum.
§ 15. MODAL LOGIC*
A. MODALITIES
Aristotle distinguishes three principal classes of premisses:
15.01 Every premiss is either about belonging to, or
necessarily belonging to, or possibly belonging to.
The expressions 'necessary' (el* avayxY)*;) and 'possible' (can
belong to, IvSe^ETai, SuvaTai) have several meanings.
1. In respect of the functor 'necessary' (or 'must') we have already
remarked (§ 13, C, 2) that it often only expresses logical consequence.
* The Aristotelian (as also the Theophrastan) modal logic is here interpreted in
the way that was customary among- the Scholastics, rediscovered by A. Becker
in 1934, and served as a basis for my ideas in works on the history of modal
logic, on Theophrastus and on ancient logic.
However, while I was writing this work, Prof. J. Lukasiewicz communicated
his being in possession of a quite different interpretation, showing the Aristote-
lian system to have contained mistakes which were rectified by Theophrastus.
This new interpretation has now been published [Aristotle's Syllogistic, 2nd ed.,
Oxford, 1957).
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THE GREEK VARIETY OF LOGIC
That this is so, can be clearly seen where for instance Aristotle
says 'It is necessary that A necessarily belongs to B' (15.02), or
again 'It is necessary that A possibly belongs to B' (15.03). The
first 'necessary' evidently means logical (a.n'k&c,) and hypothetical
(toutcov ovtcov) necessity (15.04). The necessity that something is
when it is (6t<xv JJ) obviously belongs to the second of these classes
(15.05).
2. Even the simple, unqualified (assertoric) 'belonging to', which
Aristotle often calls 'mere belonging to', is divided into an absolute
(a.Tzk&Q) and temporally qualified (xorra ^povov) kind, with different
logical properties (15.06).
3. As to possibility, Aristotle distinguishes at first two kinds:
the one-sided and the two-sided.
This distinction emerges from a searching discussion which
Aristotle conducts in the Hermeneia. The passage is of great impor-
tance for the understanding of the whole doctrine of modalities,
and so we give it in full :
15.07 When it is necessary that a thing should be, it is
possible that it should be. (For if not, the opposite follows,
since one or the other must follow; so, if it is not possible, it
is impossible, and it is thus impossible that a thing should be
which must necessarily be; which is absurd.) Yet from the
proposition 'it may be' it follows that it is not impossible, and
from that it follows that it is not necessary; it comes about
therefore that the thing which must necessarily be need not
be; which is absurd. But again, the proposition 'it is necessary
that it should be' does not follow from the proposition 'it
may be', nor does the proposition 'it is necessary that it
should not be'. . . . For if a thing may be, it may also not be,
but if it is necessary that it should be or that it should not be,
one of the two alternatives will be excluded. It remains,
therefore, that the proposition 'it is not necessary that it
should not be' follows from the proposition 'it may be'. For
this is true also of that which must necessarily be.
The sequence of thought here is, in summary form: If something
is necessary, then it is also possible; but what is possible, can also
not be (it is not impossible that it should not be) ; but from that it
follows that it is not necessary, and so a contradiction results. The
solution consists in distinguishing the two meanings of being
'possible':
15.071 One-sided possibility: that is possible which not
necessarily is not (which is not impossible).
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ARISTOTLE
15.072 Two-sided possibility: that is possible which neither
necessarily is nor necessarily is not (nor impossibly is).
This second, two-sided possibility is the one intended in the
syllogistic, and Aristotle only uses the first kind when forced to it.
He defines two-sided possibility in the Prior Analytics thus:
15.08 I use the terms 'to be possible' and 'the possible' of
that which is not necessary but, being assumed, results in
nothing impossible.
It coincides, as can be seen, with the definition just fdven above;
we find a similar one in the Metaphysics (15.09).
In two texts - but both extremely unclear and so hard to reconcile
with the teaching as a whole as to constitute an unsolved problem
(15.10) - Aristotle subdivides two-sided possibility. In the first
passage he speaks of a possibility in the sense of 'in most cases'
(&$ £7il to 7U>Xu) and of another besides (15.11); in the second
passage he distinguishes between a 'natural' (to TC(pi)XO£ \jtA^/zvj
and an indeterminate (to aopiaTOv) or 'contingent' (to arco t's/y^
possibility which, so he says, is no concern of science. Both passages
are probably interpolations.
B. STRUCTURE OF MODAL SENTENCES
The normal use of 'possible' in the sense of two-sided possibility
is a distinguishing characteristic of Aristotle's modal logic. Another,
of no less importance, is his view of the structure of modal sentences.
He only gives explicit expression to this view in one place, but it
lies at the base of the whole modal syllogistic and exercises a most
remarkable influence.
15.13 The expression 'it is possible for this to belong to
that' may be understood in two senses: either as 'to the thing
to which that belongs' or as 'to the thing to which that can
belong' ; for 'to that of which B (is predicated) A can (belong)'
means one of the two: 'to that of which B is predicated' or
'to that of which (B as) possibly (belonging) is predicated'.
This contains two points:
First a sentence of the form 'A belongs to B' is paraphrased by the
formula 'to that to which B belongs (of which B is predicated) A
also belongs' : implying a very subtle analysis of the sentence,
reminiscent of the modern formal implication, which we find else-
where in the Analytics (cf. 14.24).
Secondly it can be gathered from this text that the modal functor
does not determine the sentence as a whole, but part of it. So that
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THE GREEK VARIETY OF LOGIC
for Aristotle a modal sentence is not to be conceived in such a sense as :
'It is possible that: A belongs to B\ The modal functor does not
precede the whole sentence but one of its arguments. This distinction
quickly becomes still clearer, for the distinction is three times made
between two possible cases:
1. to that to which B belongs, A also can belong,
2. to that to which B can belong, A also can belong. In the
first case the modal functor determines only the consequent, in the
second case it determines the antecedent too.
This analysis is not expressly extended to necessity, but that
extension must be supposed : for otherwise many syllogisms would be
invalid.
C. NEGATION AND CONVERSION
In the Hermeneia Aristotle establishes a 'logical' square for senten-
ces with modal functors, in which the two expressions for 'possible'
(Suvoctov and evSe^ofjisvov) mean one-sided possibility. This square
can be compressed into the following scheme, in which all expressions
in any one row are equivalent:
possible not impossible not necessary not
not possible impossible necessary not
possible not not impossible not not necessary
not possible not impossible not necessary (15.14)
More complicated is the doctrine of the negation of sentences
containing the functor of two-sided possibility. Since this has been
defined by a conjunction of two sentences, Aristotle rightly deduces,
on the basis of the so-called de Morgan law (not to be found in
him) :
15.15 If anyone then should claim that because it is not
possible for C to belong to all D, it necessarily does not belong
to some D, he would make a false assumption: for it does
belong to all D, but because in some cases it belongs neces-
sarily, therefore we say that it is not possible for it to belong
to all. Hence both the propositions lA necessarily belongs to
some B' and 'A necessarily does not belong to some B: are
opposed to the proposition lA may belong to all B\
The passage is not quite clear; but the author's intention can
be formulated :
15.151 p is not possible, if and only if, one of the two, p and
not-p, is necessary.
From this it results that the negation of such a sentence issues as
an alternation, such as is in no case permissible as a premiss in an
Aristotelian syllogism. This prevents Aristotle from using reduction
to the impossible in certain cases.
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ARISTOTLE
Another result which Aristotle subtly deduces from his supposi-
tions is his doctrine of the equivalence of affirmative and negative
sentences when they contain the functor under consideration :
15.16 It results that all premisses in the mode of possibility
are convertible into one another. I mean not that the affirma-
tive are convertible into the negative, but that those which
are affirmative in form admit of conversion by opposition,
e.g. 'it is possible to belong' may be converted into 'it is
possible not to belong'; 'it is possible to belong to all' into 'it
is possible to belong to none' and 'not to all'; 'it is possible
to belong to some' into 'to some it is possible not to belong'.
And similarly in other cases.
Take the three modal sentences:
(a) 'A possibly belongs to B\
(b) 'A does not possibly belong to B\
(c) lA possibly does not belong to B\
(b) is the proper denial of (a), (c) is no denial of (a) but it is
'negative in form'. Then it is stated that sentences such as (a)
imply those such as (c), and are even equivalent to them. So we
have following laws :
15.161 p is possible if and only if p is not possible.
15.162 It is possible that A belongs to all B, if and only if,
it is possible that A belongs to no B.
15.163 It is possible that A belongs to some B, if and only
if, it is possible that to some B A does not belong.
Law^ analogous to those for ordinary conversion (14.10) hold
for sentences containing the functors of necessity and one-sided
possibility (15.17), just parallel to the corresponding laws in asser-
toric logic (15.18).
By contrast, the laws of conversion for sentences with the two-
sided functor are different: the universal negative cannot be con-
verted (15.19), but the particular negative can (15.20). The affir-
mative sentences are converted like assertoric ones (15.21).
D. SYLLOGISMS
On this basis and with the aid of the same procedures developed
for the assertoric syllogistic, Aristotle now builds the vast structure
of his system of syllogisms with modal premisses. Vast it is even
in the number of formulae explicitly considered, they are not fewer
than one hundred and thirty seven. But it appears much vaster - in
spite of many points where it is incomplete - in view of the subtlety
85
THE GREEK VARIETY OF LOGIC
with which the original master-logician operates in so difficult a
field. De modalibus non gustabit asinus was a medieval proverb;
but one does not need to be a donkey to get lost in this maze of
abstract laws: Theophrastus quite misunderstood the system, and
nearly all the moderns, until 1934.
The syllogisms which it comprises can be arranged in eight
groups. If we write 'A7' for a premiss with the functor 'necessary',
'Af for one with the functor 'possible' and 'A' for an assertoric
premiss (that is, one which predicates mere belonging to), these
groups can be shown as follows:
Group 12 3 4 5 6 7 8
Major premiss N N A M M A M N
Minor premiss N A N M A M N M
An.Pr.A, chap. 8 9-11 9-11 14/17/20 15/18/21 16/19/22
A striking characteristic of this syllogistic is that in very many
syllogisms the conclusion (contrary to 14.28) has a stronger
modality than the premisses, necessity being reckoned as stronger
that mere belonging to and this as stronger than possibility.
15.22 It happens sometimes also that when one premiss is
necessary the conclusion is necessary, not however, wThen
either premiss is necessary, but only when the major is, e.g.
if A is taken as necessarily belonging or not belonging to B,
but B is taken as simply belonging to C: for if the premisses
are taken in this way, A will necessarily belong or not belong
to C. For since A necessarily belongs, or does not belong, to
every B, and since C is one of the J5s, it is clear that for C
also the positive or the negative relation to A will hold
necessarily.
And of course that is the case, if one presupposes the structure
of the modal sentences as given above. For then the syllogism here
described (an analogue of Barbara) will be interpreted as follows:
If to all to which B belongs, A necessarily belongs,
and to that to which C belongs, B belongs,
then to all to which C belongs, A necessarily belongs,
which is clearly correct.
Hence it is quite wrong to extend the validity of the principle
'the conclusion follows the weaker premiss' (cf. 14.28 and 17.17)
to Aristotle's modal syllogistic.
Another striking fact is that there are numerous valid modal
syllogisms whose analogues in the assertoric syllogistic are invalid,
as for instance those two negative premisses (in opposition to
14.25) ; this is especially the case when the modal syllogism has a
premiss with the functor of possibility where the assertoric analogue
has an affirmative. For, as has been said, affirmative and negative
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ARISTOTLE
possible premisses are equivalent and can replace one another.
We take as an example a passage where Aristotle, after giving an
analogue of Barbara, in the fourth group, to which he refers by
such phrases as 'previously', 'the same syllogism', 'as before', then
proceeds:
15.23 Whenever A may belong to all B, and B may belong
to no 67, then indeed no syllogism results from the premisses
assumed, but if the premiss BC is converted after the manner
of problematic propositions, the same syllogism results as
before. For since it is possible that B should belong to no C,
it is possible also that it should belong to all C. This has been
stated above. Consequently if B is possible for all C, and A
is possible for all B, the same syllogism again results. Similarly
if in both the premisses the negative is joined with 'it is
possible': e.g. if A may belong to none of the Bs, and B to
none of the Cs. No syllogism results from the assumed premis-
ses, but if they are converted we shall have the same syllogism
as before.
This syllogistic is, like the assertoric, axiomatized. There serve
as axioms the syllogisms of the first figure in all groups, except the
sixth and eighth, together with the laws of conversion and, when
assertoric premisses occur, principles of the assertoric syllogistic. The
other syllogisms are reduced to those axioms, mostly by conversion
of premisses (direct procedure). Reduction to the impossible serves
to prove syllogisms of the first figure in the eighth group and the
analogue of Bocardo in the fifth. The analogues of Baroco and
Bocardo in the first group are proved only by ecthesis, while the
same analogues in the second and third groups remain unproved,
though it should not be hard to prove them.
The hardest problem for Aristotle are the syllogisms of the sixth
group. The first figure ones among them rightly do not rank as
intuitively evident; e.g. the analogue of Barbara would be:
If to all to which B belongs A belongs,
and to all to which C belongs, B may belong,
then to all to which C belongs, A belongs.
For this to be evident one would have to see the Tightness of the
sentence 'To all to which B may belong, B belongs' ; but according
to the definition of possibility, that is false. The details of Aristotle's
complicated attempts to validate this syllogism are matter of
conjecture and dispute, but the fact that he has to replace the
problematic minor premiss with an assertoric one is a sufficient
indication of its essential weakness (15.24). However, the passage
which contains this 'proof is one of the few where Aristotle rises to
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THE GREEK VARIETY OF LOGIC
the use of propositional variables, and for that reason remains of the
greatest logical interest (15.25).
This abortive proof is moreover not the only inconsistency in the
Aristotelian modal logic. There are for instance essential difficulties
in connection with the conversion of premisses with the functor of
necessity, and consequently in the proving of many syllogisms which
contain such premisses. In general one gets the impression that this
modal logic, by contrast to the assertoric syllogistic, is still only
in a preliminary and incomplete stage of development.
16. NON-ANALYTIC LAWS AND RULES
For those reasons there is no possible doubt that the theory of
what Aristotle would have called 'analytic' syllogisms is his chief ac-
complishment in the field of formal logic. And so great an accomplish-
ment is it from the historical and systematic points of view, that
later, 'classical' logicians have mostly overlooked all else in his
work. Yet the Organon contains a profusion of laws and rules of
other kinds. Aristotle himself recognized some of them as autono-
mous formulae, irreducible to his syllogistic. In other words: he
saw that a 'reduction' of these laws and rules to the syllogistic is
impossible - a thing which all too many after him did not see.
From the historical standpoint these formulae are to be divided
into three classes : first we have the formulae which are to be attri-
buted to a period in which Aristotle had not yet discovered his
analytic syllogisms. These are to be found in the Topics (and in the
Rhetoric). Some of them were later re-edited with the help of vari-
ables, and recognized as valid also in the period of the Analytics.
Secondly there are the formulae which Aristotle indeed considered
but mistakenly, as analytic, the syllogismi obliqui as they were later
called. Finally, in reviewing the completed system of his syllogistic
he discovered the 'hypothetical' procedure and in some cases
attained to full consciousness of propositional formulae.
But all these formulae are contained only in asides, and were
never systematically developed as was the syllogistic. Furthermore,
Aristotle thought, quite rightly in view of his methodological
standpoint, that only the analytic formulae were genuinely 'scienti-
fic', i.e. usable in demonstration.
We give first the passages which, as it seems, introduce us to
Aristotle's last thoughts on this question, then the actual non-
analytic formulae divided into five classes: those belonging to the
logic of classes, to the theory of identity, to the 'hypothetical'
syllogistic, to the theory of relations, and to propositional logic.
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ARISTOTLE
A. TWO KINDS OF INFERENCE
16.01 In some arguments it is easy to see what is wanting,
but some escape us, and appear to be syllogisms, because
something necessary results from what has been laid down,
e.g. if the assumptions were made that substance is not
annihilated by the annihilation of what is not substance, and
that if the elements out of which a thing is made are annihi-
lated, then that which is made out of them is destroyed : these
propositions being laid down, it is necessary that any part
of substance is substance; this has not, however, been drawn
by syllogism from the propositions assumed, but premisses
are wanting. Again if it is necessary that animal should
exist, if man does, and that substance should exist if man
does: but as yet the conclusion has not been drawn syllo-
gistically: for the premisses are not in the shape we required.
We are deceived in such cases because something necessary
results from what is assumed, since the syllogism also is
necessary. But that which is necessary is wider than the
syllogism : for every syllogism is necessary, but not every-
thing which is necessary is a syllogism.
We must pass over the first example, about parts of substance, as
its elucidation would take up too much space. But the second is
clear; it concerns a law, not of propositional, but of predicate logic:
16.011 If, when x is A then it is B, and when x is B then
it is C, then, when x is A then it is C.
This is a correct logical formula, and Aristotle is quite right in
saying that it permits necessary inference. Hence he also realized
that it falls under his definition of syllogism (10.05). But he refuses
to admit it as syllogism. That means that his conception of syllo-
gism had developed between the time when he penned the definition
and that when he penned this passage. The definition applies to all
correct logical formulae (and substitutions in them), but only a
sub-class retains the name 'syllogism'. We know what this sub-class
is that of the 'analytic' syllogisms. All other formulae may indeed be
logically necessary, but are not genuine syllogisms.
This distinction is not merely a matter of terminology. That
becomes evident in the passages where Aristotle deals with the
'hypothetical' syllogisms.
16.02 We must not try to reduce hypothetical syllogisms;
for with the given premisses it is not possible to reduce them.
For they have not been proved by syllogism, but assented to
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THE GREEK VARIETY OF LOGIC
by agreement. For instance if a man should suppose that unless
there is one faculty of contraries, there cannot be one science,
and should then argue that there is no* (one) faculty of
contraries, e.g. of what is healthy and what is sickly: for the
same thing will then be at the same time healthy and sickly.
He has shown that there is not one faculty of all contraries,
but he has not proved that there is not a science. And yet
one must agree. But the agreement does not come from a
syllogism, but from an hypothesis. This argument cannot be
reduced : but the proof that there is not a single faculty can.
Evidently a substitution is there being made in the law:
If (1) when not p then not q, and (2) not p, then (3) not q. (2) is
proved by an analytic syllogism, but as (1) is merely supposed and
not proved, the conclusion (3) also counts as not proved. That may
be so, but Aristotle has not noticed that the assumed formula is no
mere supposition but a correct logical law. The matter is still worse
in the next text, an immediate continuation of the last:
16.03 The same holds good of arguments which are brought
to a conclusion per impossibile. These cannot be analyzed
either; but the reduction to what is impossible can be analyzed
since it is proved by syllogism, though the rest of the argument
cannot, because the conclusion is reached from an hypothesis.
But these differ from previous arguments : for in the former a
preliminary agreement must be reached if one is to accept
the conclusion; e.g. an agreement that if there is proved to
be one faculty of contraries, then contraries fall under the
same science; whereas in the latter, even if no preliminary
agreement has been made, men still accept the reasoning,
because the falsity is patent, e.g. the falsity of what follows
from the assumption that the diagonal is commensurate, viz.
that then odd numbers are equal to evens.
So in reduction to the impossible too, Aristotle regards the
inference as not 'demonstrated', though he has to recognize that no
agreement needs to be presupposed to warrant inference.
One could express this doctrine as follows: the class of correct
formulae contains two sub-classes : that of the 'better' and that of the
'less good' in relation to 'scientific demonstration'. The less good
* Reading uia with the manuscript tradition A2B2C2T, against the (better)
tradition ABCnAl, Waitz and Ross. For raccra, read by the latter, would mean
a logical mistake in Aristotle which seems to me unlikely in this connection. For
the critical apparatus vid. Ross ad loc.
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ARISTOTLE
are precisely our non-analytic formulae, for which we have chosen
this name because according to Aristotle they are not reducible to
the classical syllogistic, 'not analyzable into the figures'. (In this lie
is evidently right, by contrast to a certain tradition.) That does not
mean that these formulae are worthless for him; on the contrary
he views them with a lively interest.
16.04 These points will be made clearer by the sequel,
when we discuss the reduction to impossibility. ... In the
other hypothetical syllogisms, I mean those which proceed
by substitution, or positing a certain quality, the inquiry will
be directed to the terms of the problem to be proved - not the
terms of the original problem, but the new terms introduced;
and the method of the inquiry will be the same as before.
But we must consider and determine in how many ways
hypothetical syllogisms are possible.
16.05 Many other arguments are brought to a conclusion
by the help of an hypothesis; these we ought to consider and
mark out clearly. We shall describe in the sequel differences,
and the various ways in which hypothetical arguments are
formed: but at present this much must be clear, that it is not
possible to resolve such arguments into the figures. And we
have explained the reason.
On that Alexander of Aphrodisias remarks:
16.06 He says that many others (syllogisms) besides are
formed from hypotheses, and promises to treat thoroughly
of them later. But no writing of his on this subject is extant.
B. LAWS OF CLASS- AND PREDICATE-LOGIC
16.07 If man (is) an animal, what is not-animal is not man.
16.08 If the pleasant (is) good, the not-good (is) not pleasant.
Notice that quantifiers are here lacking: so it is not a question of
contraposition in the ordinary sense of the word.
Aristotle was well aware that conversion of such sentences is
invalid :
16.09 For animal follows on man, but not-animal does not
(follow) on not-man; the reverse is the case.
Here there belong perhaps some rules which otherwise interpreted
could be counted in with those of the 'logical square' :
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THE GREEK VARIETY OF LOGIC
16.10 When we have shown that a predicate belongs in
every case, we shall also have shown that it belongs in some
cases. Likewise, also, if we show that it does not belong in
any case, we shall also have shown that it does not belong in
every case.
It is to be noted here that these formulae are not laws but, as is
mostly the case in the Topics, rules.
A series of similar laws is concerned with contrariety (in the sense
of the earlier notion: 12.02):
16.11 Health follows upon vigour, but disease does not
follow upon debility; rather debility follows upon disease.
16.12 Public opinion grants alike the claim that if all
pleasure be good, then also all pain is evil, and the claim that
if some pleasure be good, then also some pain is evil. More-
over, if some- form of sensation be not a capacity, then also
some form of failure of sensation is not a failure of capa-
city. . . . Again, if what is unjust be in some cases good, then
also what is just is in some cases evil; and if what happens
justly is in some cases evil, then also what happens unjustly
is in some cases good.
It may be doubted whether Aristotle continued to recognize these
laws as valid. But they are not without historical and even syste-
matic interest.
C. THEORY OF IDENTITY
As already noted (11.11) Aristotle distinguishes three kinds of
identity. Concerning the first, numerical identity, he developed
the outline of a theory; its discovery is often falsely attributed to
Leibniz.
16.13 Again, look and see if, supposing the one to be the
same as something, the other also is the same as it: for if
they be not both the same as the same thing, clearly neither
are they the same as one another. Moreover, examine them in
the light of their accidents or of the things of which they are
accidents: for any accident belonging to the one must belong
also to the other, and if the one belongs to anything as an
accident, so must the other also. If in any of these respects
there is a discrepancy, clearly they are not the same.
We have here in very compressed form a rather highly developed
doctrine of identity; indeed this text contains a greater number of
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ARISTOTLE
fundamental laws of identity than the corresponding chapter of
Principia Malhemalica (*13), and moreover Aristotle was the first
to call to mind identity, in the passage mentioned above (\2AHt. The
laws here sketched, can be formulated as follows with the help of
variables:
16.131 If B is identical with A, and C is not identical with
A, then B and C are not identical.
16.132 If A and B are identical, then (for all C): if Cbelongs
to A, then it belongs also to B.
16.133 If A and B are identical, then (for all C) : if A belongs
to C, then it belongs also to B.
16.134 If there is a C which belongs to A but not to B, then
A and B are not identical.
16.135 If there is a C to which A belongs but B does not,
then A and B are not identical.
Admittedly the last two laws are only hinted at. In another
passage we find :
16.14 For only to things that are indistinguishable and one
in essence is it generally agreed that all the same attributes
belong.
This is almost the Leibnizian principium indiscernibilium in so
many words, originating as we see with Aristotle. It is remarkable
that we do not find the simple principle :
16.141 If A is identical with B, and B with C, then A is
identical with C.
D. SYLLOGISMS FROM HYPOTHESES
Aristotle did not know the expression 'hypothetical syllogism',
but he often speaks of syllogisms from hypotheses (si* \)~o$£gz(x>c,) .
We have shown above (in the commentary on citations 16.02 and
16.03) that in general these need not be hypotheses; usually it is a
matter only of logical laws or rules, similar to syllogisms in certain
respects but not reducible to them. We have already seen some
examples of such formulae. Here are some more which Aristotle
would probably class with them.
16.15 The refutation which depends upon the consequent
arises because people suppose that the relation of consequence
is convertible. For whenever, if this is, that necessarily is, they
suppose that also when that is, this is.
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THE GREEK VARIETY OF LOGIC
16.16 When two things are so related to one another, that
if the one is, the other necessarily is, then if the latter is not,
the former will not be either, but if the latter is, it is not
necessary that the former should be.
16.17 If this follows that, it is claimed that the opposite
of this will follow the opposite of that. . . . But that is not so;
for the sequence is vice versa.
16.18 In regard to subjects which must have one and one
only of two predicates, as (e.g.) a man must have either
disease or health, supposing we are well supplied as regards
the one for arguing its presence or absence, we shall be well
equipped as regards the remaining one as well. This rule is
convertible for both purposes : for when we have shown that
the one attribute belongs, we shall have shown that the
remaining one does not belong; while if we show that the one
does not belong, we shall have shown that the remaining one
does belong.
We have there, evidently, the exclusive alternative (negation of
equivalence).
16.19 In general whenever A and B are such that they
cannot belong at the same time to the same thing, and one
of the two necessarily belongs to everything, and again C and
D are related in the same way, and A follows C but the
relation cannot be reversed, then D must follow B and the
relation cannot be reversed. And A and D may belong to the
same thing, but B and C cannot. First it is clear from the
following consideration that D follows B. For since either C
or D necessarily belongs to everything; and since C cannot
belong to that to which B belongs, because it carries A along
with it and A and B cannot belong to the same thing; it is
clear that D must follow B. Again since C does not reciprocate
with A, but C or D belongs to everything, it is possible that A
and D should belong to the same thing. But B and C cannot
belong to the same thing, because A follows C; and so some-
thing impossible results. It is clear then that B does not
reciprocate with D either, since it is possible that D and A
should belong at the same time to the same thing.
This text is one of the peaks of Aristotelian logic: the founder of
our science conducts himself with the same sureness and freedom
as in the best parts of his syllogistic, though here dealing with a new
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ARISTOTLE
field, that of non-analytic formulae. The run of this text can be
formulated thus:
(1) For all X: A or B (and not both) belongs to X, and
(2) for all X: C or D (and not both) belongs to X, and
(3) for all X: if C belongs to X, then it belongs also to A.
From these hypotheses there follows on the one hand :
(4) for all X: if B belongs to X, then it belongs also to D,
and on the other:
(5) for all X: not both B and C belong to X.
These consequences are quite correct. The thing to notice is that
there are three different binary propositional functors ('or', 'and', 'if...
then'). In his proof Aristotle uses, among others, the following three
laws, apparently with full consciousness:
16.191 For all X: if not both A and B belong to X, and B
belongs to X, then A does not belong to it.
16.192 For all X : if, when A belongs to X B also belongs to
X, but B does not belong to X, then A also does not belong
toX
16.193 For all X: if either A or B belongs to X, and A does
not belong to it, then B does belong to it.
E. LAWS OF THE LOGIC OF RELATIONS
16.20 If knowledge be a conceiving, then also the object
of knowledge is an object of conception.
16.21 If the object of conception is in some cases an object
of knowledge, then also some form of conceiving is knowledge.
16.22 If pleasure is good, then too a greater pleasure is a
greater good ; and if injustice is bad, then too a greater injustice
is a greater evil.
In this connection the following piece of history deserves to go on
record. De Morgan stated that the whole Aristotelian logic was
unable to prove that if the horse is an animal, then the head of the
horse is head of an animal. The reproach is evidently unjustified,
since the law stated in 16.20 is just what is needed for this proof.
Further, Whitehead and Russell (16.23) remark that the supposed
lack of this law is really a good point abou£ Aristotelian logic, since
it is invalid without an additional existential postulate. This may
be right in relation to De Morgan's problem, i.e. if he understood
'horse' as an individual name; but the law in which 16.20 is a
substitution is correct - since it concerns not an individual but a
class name ('knowledge').
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THE GREEK VARIETY OF LOGIC
Aristotle gives three further laws of the logic of relations in the
chapter about those syllogisms which later came to be called
'obliqui' :
16.24 That the first term belongs to the middle, and the
middle to the extreme, must not be understood in the sense
that they can always be predicated of one another. . . . But
we must suppose the verb 'to belong' to have as many
meanings as the senses in which the verb 'to be' is used, and
in which the assertion that a thing 'is' may be said to be true.
Take for example the statement that there is a single science
of contraries. Let A stand for 'there being a single science',
and B for things which are contrary to one another. Then A
belongs to jB, not in the sense that contraries are the fact of
there being a single science of them, but in the sense that it is
true to say of the contraries that there is a single science of
them.
16.25 It happens sometimes that the first term is stated of
the middle, but the middle is not stated of the third term, e.g.
if wisdom is knowledge, and wisdom is of the good, the con-
clusion is that there is knowledge of the good. The good then
is not knowledge, though wisdom is knowledge.
16.26 Sometimes the middle term is stated of the third, but
the first is not stated of the middle, e.g. if there is a science of
everything that has a quality, or is a contrary, and the good
both is a contrary and has a quality, the conclusion is that
there is a science of the good, but the good is not science, nor
is that which has a quality or is a contrary, though the good is
both of these.
16.27 Sometimes neither the first term is stated of the
middle, nor the middle of the third, while the first is sometimes
stated of the third, and sometimes not: e.g. if there is a genus
of that of which there is a science, and if there is a science of
the good, we conclude that there is a genus of the good. But
nothing is (there) predicated of anything. And if that of which
there is a science is a genus, and if there is a science of the good,
we conclude that the good is a genus. The first term then is
predicated of the extreme, but in the premisses one thing is
not stated of another.
We have here four more relational laws ; of greater importance is the
introductory remark that the so-called 'copula' need not be 'is' but
can be replaced by some other relation. A further interesting fact is
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ARISTOTLE
that Aristotle presupposes inter alia the following law from the logic
of classes (in 16.26):
16.261 For all x: if x is A and /J, then x is A or B.
The introductory remark admittedly only reveals an intuition
that is undeveloped. Nor did Aristotle link it up with his own pene-
trating thesis of the manifold structure of the sentence according to
the diversity of the categories (11.15), so rising to a higher syste-
matic unity. Nevertheless the text cited does contain the beginnings
of a logic of relations.
Finally we can collect from at least six places in the Topics a
group of rules, totalling eighteen altogether, which perhaps are
to be interpreted as belonging to the logic of relations. We give three
of them, again concerned with 'more' :
16.28 Moreover, argue from greater and less degrees. In
regard to greater degrees there are four commonplace rules.
One is : See whether a greater degree of the predicate follows a
greater degree of the subject: e.g. if pleasure be good, see
whether also a greater pleasure be a greater good. . . . Another
rule is: If one predicate be attributed to two subjects; then
supposing it does not belong to the subject to which it is the
more likely to belong, neither does it belong where it is less
likely to belong; while if it does belong where it is less likely
to belong, then it belongs as well where it is more likely. . . .
Moreover: If two predicates be attributed to two subjects,
then if the one which is more usually thought to belong to the
one subject does not belong, neither does the remaining
predicate belong to the remaining subject; or. if the one which
is less usually thought to belong to the one subject does be-
long, so too does the remaining predicate to the remaining
subject.
F. PROPOSITIONAL RULES AND LAWS
Finally we find in Aristotle four formulae belonging to the most
abstract part of logic, namely, propositional logic. Two of them even
contain propositional variables:
16.29 If when A is, B must be, (then) when B is not. A
cannot be.
That these are propositional variables, Aristotle says expressly:
16.30 A is posited as one thing, being two premisses taken
together.
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THE GREEK VARIETY OF LOGIC
16.31 If, when A is, B must be, (then) also when A is
possible, B must be possible.
It is to be noted that these propositional variables permit sub-
stitution only of quite determinate expressions, namely conjunctions
of premisses suitable for an analytic syllogism.
16.32 From true premisses it is not possible to draw a false
conclusion, but a true conclusion may follow from false
premisses, true, however, only in respect to the fact, not to the
reason.
That is not yet the scholastic principle ex falso sequitur quodlibel,
but only the assertion that one can form syllogisms in which one or
both premisses are false, the conclusion true.
16.33 If the conclusion is false, the premisses of the argument
(Xoyo?) must be false, either all or some of them.
This rule underlies the indirect proof of syllogisms (cf. 14.201-202).
Note that it is a rule, not a law, and formulated quite generally,
without being limited to two premisses.
SUMMARY
Reviewing the logical doctrines of Aristotle as presented, we can
state :
1 . Aristotle created formal logic. For the first time in history we
find in him: (a) a clear idea of universally valid logic law, though he
never gave a definition of it, (b) the use of variables, (c) sentential
forms which besides variables contain only logical constants.
2. Aristotle constructed the first system of formal logic that we
know. This consists exclusively of logical laws, and was developed
axiomatically, even in more than one way.
3. Aristotle's masterpiece in formal logic is his syllogistic. This is a
system of term-logic consisting of laws, not rules. In spite of certain
weaknesses it constitutes a faultlessly constructed system.
4. Besides the syllogistic, Aristotle constructed other portions of
term-logic, including an extremely complex modal logic, as well as a
number of laws and rules which overstep the bounds of the syllo-
gistic.
5. At the end of his life Aristotle, in a few texts, succeeded in
formulating even propositional formulae; but these, like the non-
analytic formulae of term-logic, he did not develop systematically.
6. Aristotelian logic, though formal, is not formalislic. It is lacking
also in understanding of the difference between rules and laws, and
the semantics remain rudimentary, in spite of the many works
which Aristotle devoted to the subject.
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THEOPHRASTUS
It is no exaggeration to say that nothing comparable h;j.s been seen
in the whole history of formal logic. Not only is Aristotle's logic,
according to all our information, a completely new creation, but it
has been brought even by him to a high degree of completeness.
Since moreover Aristotle's most important writings - most impor-
tant because they were the only complete logical works - survived
the cultural catastrophe of Greece, it is no wonder that the huge body
of doctrine they contain should have continued to fascinate nearly
all logicians for more than two thousand years, and that the whole
history of logic has developed along the lines traced out in advance
by Aristotle's thought.
That has not been harmless for the development of our science.
Even in antiquity there was a school of logicians which introduced a
new set of problems different from those posed by the logic of Ari-
stotle. We have only fragments of their work, and the authority of
the founder of logic was so great that the achievements of this
school were not at all understood during the long period from the time
of the Renaissance up to and including the nineteenth century. We
must now concern ourselves with them, but first a brief word must
be said about Aristotle's first disciple, Theophrastus.
§17. THEOPHRASTUS
Theophrastus of Eresos, Aristotle's chief disciple and leader of the
Peripatetic school after the founder's death, has, in company with
his less significant colleague Eudemus, an important place in the
history of logic, and that in three respects. First, he developed various
of his master's doctrines in such a way as to prepare the ground for
the later 'classical' logic; secondly, he set his own quite different
system in opposition to the Aristotelian modal iogic; thirdly, he
developed a doctrine of hypothetical arguments which was a pre-
paration for Megarian-Stoic logic.
His very numerous works (17.01) have all perished save for some
one hundred fragments. These, however, are enough to tell us that he
commented on the most important of Aristotle's logical works ( 1 7.02 1 ,
and they give us some insight into his own logical thought.
A. DEVELOPMENT AND ALTERATION OF VARIOUS DOCTRINES
17.03 Speech having a twofold relation - as the philoso-
pher Theophrastus has shown - one to the hearers, to whom it
signifies something, the other to the things about which it
informs the hearers, there arise in respect of the relation to the
hearers poetics and rhetoric, ... in respect of that to the things,
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THE GREEK VARIETY OF LOGIC
it will be primarily the philosopher's business, as he refutes
falsehood and demonstrates truth.
We can see that this is a new semiotic, with stress on what is now
called the 'pragmatic' dimension of signs.
17.04 Theophrastus rightly calls the singular sentence
determined, the particular undetermined.
17.05 Alexander opines that 'not belonging to all' and 'to
some not belonging' differ only in the expression, whereas
Theophrastus's view is that they differ also in meaning: for
'not belonging to all' shows that (something) belongs to
several, 'to some not belonging' that (not belonging) to one.
A more important thought is the following:
17.06 Consequently Theophrastus says that in some cases,
if the determination (of quantity) TupoaSiopLCT^o^) does not
also stand with the predicate, opposites, contradictories, will
be true, e.g. he says that 'Phanias possesses knowledge',
'Phanias does not possess knowledge', can both be true.
This is not a matter, as Theophrastus mistakenly supposed, of
quantification of the predicate, which Aristotle had rejected (12.03),
but of a quantification of both parts of a subject when there is a
two-place functor (cf. 44.22 ff.). This structure was only later treated
in detail (cf. 28.15 ff., 42.06, 42.22). We have here the first beginnings
of it.
17.07 In those premisses which potentially contain three
terms, viz. those . . . which Theophrastus called xa-ra 7rpocrAY)tJnv
(for these have three terms in a sense; since in (the premiss) 'to
all of that to all of which B belongs, A also belongs' in the two
terms A and B which are explicit there is somehow comprised
the third of which B is predicated . . .): (these premisses)
. . . seem to differ from categorical ones only in expression, as
Theophrastus showed in his On Affirmation.
17.08 But Theophrastus in On Affirmation treats 'to that
to which B (belongs, there belongs also) A' as equivalent
(feov Suva(X£V7]v) to 'to all of that to all of which B belongs,
A (also belongs) (cf. 14.24).
17.09 But Theophrastus and Eudemus have given a
simpler proof that universal negative premisses can be con-
verted. . . . They conduct the proof so: A belongs to no B.
If it belongs to none, A must be disjoined (dbus^suxToci) and
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THEOPHRASTUS
separated (xs/wptdTat) from B. But what is disjoined is
disjoined from something disjoined. Therefore B too is quite
disjoined from A. And if this is so, it belongs to no A.
This shows that Theophrastus takes a purely extensional view of
the terms (cf. §36, E) - so much so that one is led by this text (as
by 17.13) to think of a diagrammatic scheme such as Leibniz used
(36.14).
17.10 To these four (Aristotelian syllogisms of the first
figure) Theophrastus added five others, which are neither
perfect nor indemonstrable.
We no longer have the relevant text. Alexander's explanations
(17.11) show that these are the five:
17.111 If A belongs to all B and B to all C, then too C
belongs to some A (Baralipton).
17.112 If A belongs to no B, and B to all C, then too C
belongs to no A (Celanles).
17.1 13 If A b longs to all B and B to some C, then too C be-
longs to some A (Dabitis).
17.114 If A belongs to all C and B to no C, then to some
A C does not belong (Fapesmo).
17.115 If A belongs to some B and B to no 67, then too to
some A C does not belong (Frisemomorum).
These are what were later called the 'indirect' syllogisms of the
first figure, deduced by means of the Aristotelian rules (cf. § 13, D).
B. MODAL LOGIC
All the texts so far quoted contain developments of or - often
questionable - improvements on the Aristotelian logic. The Theo-
phrastan theory of modal syllogisms, on the other hand, is an
entirely new system, set in fundamental oppositions, as it appears,
to that of Aristotle.
17.12 Hence Aristotle says that universal negative possible
premisses are not convertible. But Theophrastus says that
these too, like the other negatives, can be converted.
17.13 But Theophrastus and Eudemus, as we have already-
explained at the beginning, say that universal negative
(possible premisses) can be converted, like universal negative
assertoric and necessary ones. Their convertibility they prove
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THE GREEK VARIETY OF LOGIC
thus: if A possibly does not belong to all B, B also possibly
does not (belong) to all A; for if A possibly does not belong
to all B . . ., then A can be disjoined from all B; but if this is
so, B also can be disjoined from A; and in that case B also
possibly does not belong to all A.
17.14 It is (according to Aristotle) a property of the
possible to convert, i.e. the affirmations and negations con-
cerning it follow on each other . . . but it should be known that
this conversion of premisses is not valid in the school of
Theophrastus, and they do not use it. For there is the same
reason (1) for saying that the universal negative possible
(premiss) is convertible, like the assertoric and necessary, and
(2) (for saying) that affirmative possibles are not convertible
into negatives.
In brief: according to Theophrastus all laws of conversion for
problematic sentences are exactly analogous to those for assertoric
sentences; and the 'reason' of which Alexander speaks, can only be,
so it would seem, that the modal doctrine of Theophrastus is based
on one-sided possibility, while Aristotle's is based on two-sided.
Similarly the second fundamental thesis of the Aristotelian system
is also rejected: for Theophrastus the functor of modality must be
thought of as determining the whole sentence, not just one or both
of its arguments, i.e. it must be thought of as standing at the
beginning of the sentence (cf. commentary on 15.13).
17.15 But his companions who are with Theophrastus and
Eudemus, deny this, and say that all formulae consisting of a
necessary and an assertoric premiss, so constituted as to be
suitable for syllogistic inference, yield an assertoric conclusion.
They take that from the (principle according to which) in all
(syllogistic) combinations the conclusion is similar to the last
and weaker premiss.
17.16 But Theophrastus, (in order to prove) that in this
combination (auprXox?)) the conclusion yielded is not necessary,
proceeds thus: Tor if B necessarily (belongs) to C and A does
not necessarily belong to B, if one disjoins the not necessary,
evidently, as B is disjoined (from A), C too will be disjoined
from A : hence does not necessarily belong to it in virtue of
the premisses.
17.17 They prove that this is so by material means (= by
substitutions) also. For they take a necessary universal
affirmative or negative as major (premiss), an assertoric
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THEOPHRASTUS
universal affirmative as minor, and show that these yield an
assertoric conclusion. Suppose that animal (belongs; to all
man necessarily, but man belongs (simply) to all in motion :
(then) animal will not necessarily belong to all in motion.
Now the basis for Aristotle's permitting the drawing of a necessary
conclusion from one necessary and one assertoric premiss, was
precisely his idea of the structure of modal sentences. Theophrastue
certainly does not reject this idea in his extant fragments, and per-
haps was not fully aware of it. But in any case all that we have of his
modal logic gives evidence of a system presupposing the rejection
of the Aristotelian structure of modal sentences.
C. HYPOTHETICAL SYLLOGISMS
We have no text of Theophrastus that contains anything of his
theory of hypothetical propositions. He seems to have treated of
them, for he distinguished the meaning of zl and iizzi (17.18). Possibly
too it was he who introduced the terminology for these propositions
which Galen aseribes to the 'old Peripatetics' (17.19). However, we
know that he developed hypothetical syllogisms:
17.20 He (Aristotle) says that many syllogisms are formed
on hypotheses. . . . Theophrastus mentions them in his
Analytics, as do Eudemus and some others of his companions.
According to Philoponus, both of them 'and also the Stoics'
wrote 'many-lined' treatises about these syllogisms (17.21). In fact,
however, the treatment of only one kind of these syllogisms is
expressly attributed to Theophrastus, that namely which consists of
'thoroughly (St' oXcov) hypothetical' syllogisms.
17.22 However, the thoroughly hypotheticals are reduced to
the three figures in another way, as Theophrastus has proved in
the first book of the Prior Analytics. A thoroughly hypothe-
tical syllogism is of this kind: If A, then B; if B, then C; if
therefore A, then C. In these the conclusion too is hypotheti-
cal; e.g. if man is, animal is; if animal is, then substance is:
if therefore man is, substance is. Now since in these too there
must be a middle term in which the premisses convene (for
otherwise here also there cannot be a conclusive link), this
middle will be positioned in three ways. For if one premiss
ends with it and the other begins with it, there will be the
first figure ; it will be in fact as though it was predicated of one
extreme, subjected to the other. ... In this way of linking one
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THE GREEK VARIETY OF LOGIC
can take also the converse of the conclusion, in such a way that
(C) is not the consequent (E7r6[xevov) but the antecedent
(y]You(ji£vov), not indeed simply, but with opposition, since
when a conclusion 'if A, then C" has been gained, there is
gained a conclusion 'if not C, then not A\
If the premisses begin differently and end similarly, the
figure will be the second, like the second (in the system) of
categorical (syllogisms). . . . e.g. If man, then animal; if
stone, then not animal; therefore if man, then not stone. . . .
If the premisses begin similarly and end differently, the
figure will be like the third . . . e.g. if A, then B ; if not A, then
C; it will follow : therefore, if not jB, then C, or, if not 67, then B.
The formulae contained in this text are presented in such a way
that from them alone it is impossible to tell whether their variables
are term- or propositional variables. However, the substitutions
show that the former is the case. Hence we have no reason to ascribe
any law of propositional logic to Theophrastus. Yet it is most
probable that in developing Aristotle's hints about 'syllogisms from
hypotheses' he prepared the way for the Megarian-Stoic doctrine.
The formulae in the text just cited are worded as rules; but we
do not know whether this wording is due to Theophrastus himself,
or to Alexander and so mediately to the Stoics.
104
III. THE MEGARIAN-STOIC SCHOOL
§ 18. HISTORICAL SURVKY
A. THINKERS AND SCHOOLS
18.01 Euclid originated from Megara on the Isthmus. . . .
He occupied himself with the writings of Parmenides; his
pupils and successors were called 'Megarians', also 'Eristics'
and later 'Dialecticians'.
18.02 The Milesian Eubulides also belongs among the
successors of Euclid; he solved many dialectical subtleties,
such as The Liar.
18.03 Eubulides was also hostile to Aristotle and made
many objections to him. Among the successors belonged
Alexinus of Elis, a most contentious man, whence he gained
the name 'Elenxinus' ('Refuter').
18.04 Among (the pupils) of Eubulides was Apollonius,
surnamed Cronus, whose pupil Diodorus, the son of Ameinias
of Iasus, was also called Cronus . . .. He too was a dialecti-
cian. . . . During his stay with Ptolemy Soter he was challenged
by Stilpo to solve some dialectical problems; but as he could
not do this immediately . . ., he left the table, wrote a treatise
on the problems propounded, and died of despondency.
18.05 Stilpo, from the Greek Megara, studied under some
pupils of Euclid; others say that he studied under Euclid
himself, and also under Thrasymachus of Corinth, the friend
of Icthyas. He surpassed the rest in inventiveness of argument
and dialectical art to such an extent that well-nigh all Greece
had their eyes on him and was fain to follow the Megarian school.
18.06 He caught in his net Crates and very many more.
Among them he captured Zeno the Phoenician.
18.07 Zeno, the son of Mnaseas or Demeas, was born at
Citium, a small Greek town on the island of Cyprus, where
Phoenicians had settled.
18.08 He was ... a pupil of Crates; some say that he also
studied under Stilpo.
18.09 He was assiduous in discussion with the dialectician
Philo and studied with him; so that Philo came to be admired
by the more youthful Zeno no less than his master Diodorus.
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THE GREEK VARIETY OF LOGIC
18.10 He also spent some time under Diodorus, . . . studying
hard at dialectics.
18.11 Kleanthes the son of Phanias was born at Assus . . .
joined Zeno . . . and remained true to his teaching.
18.12 Chrysippus the son of Apollonius from Soli or
Tarsus . . ., was a pupil of Cleanthes.
18.13 He became so famous as a dialectician, that it was
generally said that if the gods were to use dialectic, it would be
none other than that of Chrysippus.
It was necessary to cite these extracts from the Lives and Opinions
of Famous Philosophers of Diogenes Laertius, in order to counter a
widespread error to the effect that there was a Stoic, but no Megarian
logic. From the passages quoted it appears unmistakably that (a)
the Megarian school antedated the Stoic, (b) the founders of the
Stoa, Zeno and Chrysippus, learned their logic from the Megarians,
Diodorus, Stilpo and Philo. And again (c) we know at least three
Megarian thinkers of importance in the history of logic - Eubulides,
Diodorus, and Philo - while only one can be named from the Stoa,
viz. Chrysippus who can lay claim to practically no basically original
doctrine, whereas each of the three Megarians conceived a definitely
original idea.
Admittedly the Megarian school seems to have died out by the
close of the 3rd century b.c, whereas the Stoa continued to flourish.
Also the adherents of the latter disseminated logic in many excellent
handbooks with the result that people, as in Galen's time, spoke only
of Stoic logic. The least that can justly be required of us is to speak
of a Megarian-Stoic logic. Possibly the basic ideas should be attri-
buted to the Megarians, their technical elaboration to the Stoics,
but this is mere conjecture.
The names and doctrinal influences recorded by Diogenes can be
conveniently summarized in the following table :
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MEGARIAN-STOIC LOGIC
Euclid of Megara, pupil of Socrates,
founder of the Megarian or 'dialectical' school
(ca. 400 b.c.)
Alexinus of Elis
called 'Elenxinus'
Eubulides
of Miletus
discoverer
of The Liar
I
Icthyas
| Thrasymachus
friend of
Icthyas
Apollonius
Cronus
.1
Diodorus
Cronus of Iasus
307 b.c. —
Philo of Megara
Stilpo of
Megara
(ca. 320 b.c.)
i
Zeno of Citium
founder of
the Stoa
ca. 300 b.c.
Cleanthes of Assus
Chrysippus of Soli
'second founder of
the Stoa'
281/78-208/05(7) b.c.
B. PROBLEMS OF LITERARY HISTORY
The conditions for investigation of the Megarian-Stoic logic
are much less favourable than those for that of the logic of Aristotle
or even Theophrastus. We have the essential works of Aristotle
entire, and in the case of Theophrastus are in possession at least of
fragments quoted by competent experts who are not absolutely
hostile to the author they cite. But for Megarian-Stoic teaching we
have to rely essentially on the refutations of Sextus Empiricus, an
inveterate opponent. As B. Mates rightly says, it is as though we
had to rely for a knowledge of R. Carnap's logic only on existen-
tialist accounts of it. Fortunately Sextus, though no friend to the
Stoics, was (in contrast to most existentialists) well acquainted with
formal logic, which he opposed from his sceptical point of view.
We can moreover control at least some of his reports by means of
other texts.
But still we have nothing but fragments. We can hardly doubt
that the material to hand suffers from many gaps: for instance
term-logic is almost completely missing, and it seems hardly likely
that it was wholly unconsidered in the Stoa.
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THE GREEK VARIETY OF LOGIC
Another problem concerns the interpretation. Even in antiquity
Stoic texts were often 'aristotelized', propositional variables taken
for term-variables etc. A similar defect characterizes all modern
historians of logic, Prantl most of all, who completely mistook the
significance of this logic. Peirce was the first to see that it was a
propositional logic, and J. Lukasiewicz did a lasting service in giving
the correct interpretation. Now there is available a scientific
monograph - a rarety in history of logic - by B. Mates. So in the
present state of research it can be stated with some certainty that
we are again in a position to understand this extremely interesting
logic.
C. ORIGIN AND NATURE
In reading the Megarian-Stoic fragments one's first impression
is that here is something different from Aristotelian logic: termino-
logy, laws, the very range of problems, all are different. In addition
we are confronted with a new technique of logic. The most striking
differences are that the Megarian-Stoic logic is firstly not a logic of
terms but of propositions, and secondly that it consists exclusively
of rules, not of laws - as does the Prior Analytics. The question at
once arises, what was the origin of this logic.
The answer is complex. First of all one cannot doubt that the
Megarians and Stoics, who as we have seen (cf. 18.03) found an only
too frequent delight in refutation, had a tendency to do everything
differently from Aristotle. Thus for example they introduce quite
new expressions even where Aristotle has developed an excellent
terminology.
Yet it should not be said that their logical thought could have
developed uninfluenced by Aristotle. On the contrary, they appear to
have developed just those ideas which are last to appear in the
Organon. We find, for instance, a more exact formulation of the
rules which Aristotle used in axiomatizing the syllogistic, and him-
self partially formulated. Nor can it be denied that they developed
his theory of 'syllogisms from hypotheses', chiefly on the basis of
the preparatory work of Theophrastus. And generally speaking
they everywhere show traces of the same spirit as Aristotle's, only
in a much sharper form, that spirit being the spirit of formalized
logic.
And that is not yet all. In many of his non-analytical formulae
Aristotle depends directly on pre-Platonic and Platonic discussions,
and this dependence is still greater in the case of the Megarian-
Stoic thinkers. It often happens that they transmute these discus-
sions from the language of term-logic into that of propositional
logic, and one can understand how they, rather than Aristotle,
came to do this on such a scale. Aristotle always remained at heart a
108
MEGARIAN-STOIC LOGIC
pupil of Plato's, looking for essences, and accordingly asking
himself the question: 'Does A belong to B?' But the Megarians start
from the pre-Platonic question: 'How can the statement p be
refuted?' Alexinus was called 'Hefuter', and all these thinkers
continued to be fundamentally refuters in their logic. Which means
that their basic problems were concerned with complete propositions,
whereas Aristotle had his attention fixed on terms. The thorough
empiricism too, to which the Stoics gave their allegiance, contributed
to this difference.
As to details, propositional logic originated with the Megarians
and Stoics, the second great contribution made by the Greeks to
logic, and just what was almost entirely missing from Aristotelian
logic. Then, as already stated, they understood formal treatment in
a formalistic way, and laid the foundations of an exact semantics
and syntax. Misunderstood for centuries, this logic deserves recogni-
tion as a very great achievement of thought.
Unfortunately no means is available for us to pursue the historical
development of Megarian-Stoic investigations; we can only consider
what we find at the end of this development, which seems to have
already come with Chrysippus. Within a hundred and fifty years
Greek logic rose with unbelievable speed to the very heights of
formalism. We now have to view these heights as already attained.
Our presentation cannot be historical; it can only proceed systemati-
cally.
§ 19. CONCEPT OF LOGIC. SEMIOTICS. MODALITIES
A. LOGIC
19.01 They (the Stoics) say there is a threefold division of
philosophical speech: one (part) is the physical, another the
ethical, the third the logical.
19.02 They compare philosophy to an animal, the logical
part corresponding to the bones and sinews, the ethical to the
fleshy parts, the physical to the soul. Or again to an egg, the
logical (part) being the outside (=the shell). ... Or again to a
fertile field. The fence then corresponds to the logical.
19.03 According to some the logical part is divided into
two sciences, rhetoric and dialectic. . . . They explain rhetoric
as the science of speaking well . . . and dialectic as the science
of right discussion in speech, by question and answer. Hence the
following definition: it is the science of the true, the false, and
of what is neither of the two.
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THE GREEK VARIETY OF LOGIC
That of course does not mean that the Stoics knew of a three-
valued logic (cf . 49.08) ; they refer only to sentences (which are true
or false) and their parts (which are neither). - The text cited expres-
ses the attitude of the Stoics to the problem of the place of logic
among the sciences : for them it is quite unmistakably a part of the
system. What more is said seems to concern a methodology of discus-
sion rather in the manner of the Aristotelian Topics (11.01). But as
we know from other fragments, it is only a consequence of the Stoic
doctrine of the principal subject-matter of logic which consists in
lecia (Xextoc). This important notion requires immediate clarifica-
tion.
B. LECTA
19.04 The Stoics say that these three are connected: the
significate (<yyj[jiaiv6[ji£vov), the sign (<n)(Jiaivov) and the thing
(tuyxocvov). The sign is the sound itself, e.g. the (sound) 'Dion',
the significate is the entity manifested by (this sign) and which
we apprehend as co-existing with our thought, (but) which
foreigners do not comprehend, although they hear the sound;
the thing is the external existent, e.g. Dion himself. Of these,
two are bodies, viz. the sound and the thing, and one imma-
terial, viz. the entity signified, the lecton, which (further) is
true or false.
19.05 They say that the lecton is what subsists according
to a rational presentation (xara cpavTacuav XoyixYjv).
19.06 Some, and above all those of the Stoa, think that
truth is distinguished in three ways from what is true, . . .
truth is a body, but what is true is immaterial; and this is
shown, they say, by the fact that what is true is a proposition
(a££<o[i.a), while a proposition is a lecton, and lecia are imma-
terial.
We have refrained from translating the Greek expression Xsxtov
which derives from Xsyeiv and literally means 'what is said', i.e.
what one means when one speaks meaningfully. The text last cited,
about truth and what is true, is to be specially noted. The former is
something psychic, and for the Stoics all such, every thought in
particular, is material. But the lecton is not a quality of the mind, or
in scholastic terminology a conceptus subjectivus. To use Frege's
language it is the sense (Sinn) of an expression, scholastically the
conceptus objectivus, what is objectively meant. In the (pseudo-)
Aristotelian Categories there is a passage (10.29) about the Xoyo? tou
7rpay(jLaTO^, which corresponds to the Stoic lecton. Only in the Stoa
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MEGARIAN-STOIC LOGIC
the leclon has become the chief subject-matter of logic and indeed
the unique subject of formal logic. That certainly jettisons the
Aristotelian neutrality of logic arid supposes a definite philosophical
standpoint. But the original philosophical intuition involved is to
be the more noticed in that very many philosophers and logicians, up
to the most recent times, have confused the leclon with psychic
images and occurrences (cf. 26.07, 36.08). That the Stoic logic is a
science of lecta is made plain by their division :
19.07 Dialectic is divided, they say, into the topic of
signiflcates and (the topic) of the sound. That of significates
is divided into the topic of conceptions, and that of the lecta
which co-exist with them: propositions, independents (lecta),
predicates, and so on . . ., arguments and moods and syllogisms,
and fallacies other than those arising from the sound and the
things. ... A topic proper to dialectic is also that already
mentioned about the sound itself.
C. SYNTAX
19.08 The elements of speech are the twenty-four letters.
But 'letter' can have three meanings: the letter (itself), the
(written) sign (^apaxr/jp) of the letter, and its name, e.g.
'alpha'. . . . Utterance (<pcoWj) is distinguished from locution
(Xe£is) in that a mere sound is utterance, but only articulated
sound is locution. Locution is distinguished from speech in
that speech is always meaningful, while what has no meaning
can be locution, e.g. 'blityri' - which is not speech.
19.09 There are five parts of speech, as Diogenes, in his
(treatise) On Utterance, and Ghrysippus say: proper names
(ovofi-a), general names (7rpo<77)Yopia), verbs (pjfxa), connections
(ouvSsctu,o?), articles. . . .
19.10 A general name is according to Diogenes a part of
speech which signifies a common quality, e.g. 'man', 'horse'.
But a proper name is a part of speech which manifests a
quality proper to one, e.g. 'Diogenes', 'Socrates'. A verb is
according to Diogenes a part of speech, which signifies an
incomposite predicate (xaTYjyopYjjjia), or as others (define it), an
indeclinable part of speech which signifies something co-
ordinated with one or more, eg. 'I write', 'I speak'. A con-
junction is an indeclinable part of speech which connects its
parts.
Ill
THE GREEK VARIETY OF LOGIC
Accounts of the division of lecla contradict one another, and are
obscure. The following scheme composed by B. Mates (19.11) may
best correspond to the original Stoic teaching:
What is said
Xextov
incomplete complete
zKknzic, auTOTsXli;
predicate subject proposition others
xaT7]YOp7](jLa titcoctk; a£ia)(j,a 7ttK7[ia
But the division of propositions is clearly and fully transmitted.
19.12 A proposition is what is true or false, or a complete
entity (Trpayfxa) assertoric by itself, e.g. 'It is day*, 'Dion walks
about'. It is called 'axiom' (a£ico[xa) from being approved
(a£iouc7&ai) or disapproved. For he who says 'it rs day' seems to
admit that it is day; and when it is day, the foregoing axiom is
true ; but when it is not (day), false. Different from one another
are axiom, question, inquiry, command, oath, wish, exhorta-
tion, address, entity similar to an axiom.
19.13 Of axioms, some are simple, some not simple, as is
said in the schools of Chrysippus, Archedemus, Athenodorus,
Antipater and Crinis. Simple are those which consist of an
axiom not repeated (fjrf) Swccpopou^evou), e.g. 'it is day'. Ones not
simple are those consisting of a repeated axiom or of more
than one axiom. An example of the former is: 'if it is day, it is
day' ; of the latter : 'if it is day, it is light' .
19.14 Of simple axioms some are definite (cbpLcjjjiva), others
indefinite, others again intermediate (fxicja). Definite are those
which are referentially expressed, e.g. 'this man walks about',
'this man sits' : (for they refer to an individual man). Indefinite
are those in which an indefinite particle holds the chief place
(xupLEuei), e.g. 'someone sits'. Intermediate are those such as:
' a man sits' or 'Socrates walks about'. . . .
19.15 Among axioms not simple is the compound (cjuvyj^fiivov
= conditional), as Chrysippus in the Dialectic and Diogenes in
the Dialectic AH say, which is compounded by means of the
implicative connective 'if ; this connective tells one that the
second follows from the first, e.g. 'if it is day, it is light'. An
inferential axiom (7rapaauvy)[i.(xevov) is, as Crinis says in the
Dialectic Art, one which begins and ends with an axiom and is
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MEGARIAN-STOIC LOGIC
compounded 7capamwJ7CTai) by means of the connective 'since1
(inei), e.g. 'since it is day, it is light'. This connective tells one
that the second follows from the first and that the first is the
case. Conjunctive (au[jwus7cXeYjjivov) is the axiom compounded by
means of a conjunctive connective, e.g. 'it is day and it is
light'. Disjunctive (Sis^euytiivov) is the axiom compounded by
means of the separative connective 'or', e.g. 'it is day or it is
night'. This connective tells one that one of the axioms is
false. Causal (amto<k<;) is the axiom compounded by means
of the connective 'because', e.g. 'because it is day, it is light'.
For it is here to be understood that the first is the cause of the
second. An axiom showing what is rather the case is one com-
pounded by means of the connective 'rather than' which
shows this and stands in the middle of the axiom, e.g. 'it is
night rather than day'.
Note in these texts that lecla, not words or psychic events are the
subject-matter throughout. Hence most translations (those e.g. of
Apelt, 19.16, and Hicks) are misleading, since they talk of connective
'words' and 'judgements'.
D. DOCTRINE OF CATEGORIES
19.17 The common genus 'what is' has nothing over it. It
is the beginning of things and everything is inferior to it. The
Stoics wanted to put another, still more principal genus above
it.
19.18 To some Stoics 'what' seems to be the prime genus;
and I will say why. In nature, they say, some things exist,
others do not. Even those which do not exist are contained in
nature, those which occur in the soul, like centaurs, giants
and anything else which acquires an image when falsely
framed in thought, though having no substance.
So according to these Stoics there is a summum genus. This is a
regression in comparison with Aristotle's subtle anticipation of a
theory of types (11.16).
19.19 But the Stoics think that the prime genera are more
limited in number (than the Aristotelian). . . . For they
introduce a fourfold division into subjects (u7rox£iji.sva), qualia
(tcoloc), things that are in a determinate way (ttco; £'x0VTa)> and
things that are somehow related to something (r.pbc, ti -co;
£XOVTa).
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THE GREEK VARIETY OF LOGIC
These four categories are not to be understood as supreme genera
(under the 'what'): That is, it is not the case that one being is a
subject, another a relation, but all the categories belong to every
being, and every category presupposes the preceding ones (19.20).
This doctrine has no great significance for logic.
E. TRUTH
Apart from the distinction already mentioned between what is
true and truth (19.06), the Stoics seem to have used the word 'true'
in at least five senses. On this Sextus says :
19.21 Some of them have located the true and the false in
the significates ( =leda), others in the sound, others again in
the operation of the mind.
As regards the truth of lecla, a further threefold distinction can be
made between:
1. truth of propositions.
2. truth of propositional forms (i.e. what it is that sentential
functions refer to). That the Megarians and Stoics attributed truth
and falsity to such propositional forms is seen in their teaching
about functors (vide infra).
3. truth of arguments (vid. 21.07).
Those are all lecla, but Sextus refers to two further kinds of truth:
4. truth of ideas (19.22).
5. truth of sentences.
According to all our information the first kind of truth was
fundamental, as presupposed in all the others. Thus for instance the
Stoics defined the truth of propositional forms by its means, with
the help of time-variables; the truth of arguments in terms of the
truth of the corresponding conditional propositions; while the truth
of ideas and sentences is similarly reducible to that of lecla, accord-
ing to what we know of the relation between them.
F. MODALITIES
Only fragments have come down to us of the very interesting
Megarian doctrine of modalities. It seems to be an attempt to reduce
necessity and possibility to simple existence by means of time-
variables, a proceeding wholly consonant with the empirical stand-
point of these thinkers. We give only the two most important
passages on the subject:
19.23 'Possible' can also be predicated of what is possible in
a 'Diodorean' sense, that is to say of what is or will be. For he
114
MEGAR IAN-STOIC LOGIC
(Diodorus) deemed possible only what either is or will be.
Since according to him it is possible that I am in Corinth, if
I am or ever shall be there; and if I were not going to be
there, it would not be possible. And it is possible that a child
should be a grammarian, if he will ever be one. To prove this,
Diodorus devised the master-argument (xupiefoov). Philo took a
similar view.
19.24 The (problem of) the master-argument seem- prob-
ably to have originated from the following considerations. As
the following three (propositions) are incompatible: (1)
Whatever is true about the past is necessary, (2) the impos-
sible does not follow from the possible which neither is nor
will be true - Diodorus, comparing this incompatibility with
the greater plausibility of the first two, inferred that nothing is
possible which neither is nor will be true.
Unfortunately that is the only really explicit text about the
celebrated master-argument of Diodorus. It fails to enable us to
survey the whole problem, because we do not know why the three
propositions should be incompatible. One thing seems clear: that
possibility was defined in the following way :
19.241 p is (now) possible if and only if p is now true or
will be true at some future time.
From a rather vague text of Boethius (19.25) we further learn
that the definitions of the other possibility-functors must be more
or less as follows :
19.242 p is (now) impossible if and only if p is not true and never
will be true.
19.243 p is (now) necessary if and only if p is true and always will
be true.
19.244 p is (now) not necessary if and only if p is not true or will
not be true at some future time.
§20. PROPOSITIONAL FUNCTORS
To the credit of the Megarian-Stoic school are some very subtle
researches into the most important propositional functors. The
thinkers of the school even succeeded in stating quite correct
truth-matrices.
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THE GREEK VARIETY OF LOGIC
A. NEGATION
20.01 Negative are said to be only those propositions to
which the negative particle is prefixed.
This text shows something to which many passages bear witness,
that the Stoics constructed their logic not merely formally, but quite
formalistically. This was blamed by Apuleius (20.02) and Galen
(20.03), who said that the Stoics were only interested in linguistic
form. But this reproach - if indeed it is one - cannot be sustained
in view of what we know of the subject-matter of Stoic logic; Stoic
formalism is concerned with words only as signs of lecta.
20.04 Among simple axioms are the negative (darocpaTLxov),
the denying (apv7)Ti.x6v), the privative ((jTepyraxov). . . . An
example of the negative is: 'it is not day'. A species of this
is the super-negative (u7repoc7rocpaT[.x6v). The super-negative is
the negation of the negative, e.g. 'not - it is not day'. This
posits 'it is day'. A denying (axiom) is one which consists of
a negative particle and a predicate, e.g. 'No-one walks about'.
Privative is one which consists of a privative particle and
what has the force of an axiom, e.g. 'this man is unfriendly
to man'.
The extant fragments do not contain a table of truth-values for
negation, but the text cited evidently contains the law of double
negation :
20.041 not-not p if and only if p (cf. 24.26).
B. IMPLICATION
The definition of implication was a matter much debated among
the Megarians and Stoics:
20.05 All dialecticians say that a connected (proposition) is
sound (uyti<;) when its consequent follows from (axoAoufrsi) its
antecedent - but they dispute about when and how it follows,
and propound rival criteria.
Even so Callimachus, librarian at Alexandria in the 2nd century
b.c, said:
20.06 The very crows on the roofs croak about what
implications are sound.
116
MEGARIAN-STOIC LOGIC
1. Philonian implication
20.07 Philo said that the connected (proposition) is true
when it is not the case that it begins with the true and ends
with the false. So according to him there are three ways in
which a true connected (proposition) is obtained, only one in
which a false. For (1) if it begins with true and ends with true,
it is true, e.g. 'if it is day, it is light' ; (2) when it begins with false
and ends with false, it is true, e.g. 'if the earth flies, the earth
has wings'; (3) similarly too that which begins with false and
ends with true, e.g. 'if the earth flies, the earth exists'. It is
false only when beginning with true, it ends with false, e.g.
'if it is day, it is night' ; since when it is day, the (proposition)
'it is day' is true - which was the antecedent ; and the (proposi-
tion) 'it is night' is false, which was the consequent.
Here some terminological explanations are required. The Stoics
called the antecedent "/jyouuxvov, the consequent Xvjyov, and moreover
had the corresponding verbs: ^yetTOu, Xyjyei, untranslatable in their
technical use. Hence we have simply translated these words accord-
ing to their ordinary sense, by 'begins' and 'ends'. The term too for
the sentences themselves (or the propositions to which they refer) has
been translated according to its everyday sense by 'connected', the
word 'conditional' having been avoided because apparently the
idea of condition was foreign to Megarian-Stoic thought.
As to the content of the passage, it gives us a perfect truth-
matrix, which can be set out in tabular form thus:
20.071
Antecedent
Consequent
Connected
proposition
true
true
true
false
false
true
false
true
true
true
false
false
It is, as we can see, the truth-value matrix for material impli-
cation, ordered otherwise than is usual nowadays (41.12; but 42.27).
The latter therefore deserves to be called 'Philonian'.
2. Diodorean implication
20.08 Diodorus says that the connected (proposition) is
true when it begins with true and neither could nor can end
with false. This runs counter to the Philonian position. For the
connected (proposition) 'if it is day, I converse' is true
according to Philo, in case it is day and I converse, since it
117
THE GREEK VARIETY OF LOGIC
begins with the true (proposition) 'it is day' and ends with the
true (proposition) 'I converse'. But according to Diodorus (it is)
false. For at a given time it can begin with the true (proposi-
tion) 'it is day' and end with the false (proposition) ' I converse' ,
suppose I should fall silent . . . (and) before I began to con-
verse it began with a true (proposition) and ended with the
false one 'I converse'. Further, the (proposition) 'if it is
night, I converse' is true according to Philo in case it is day
and I am silent; for it (then) begins with false and ends with
false. But according to Diodorus (it is) false; for it can begin
with true and end with false, in case the night is past and
I am not conversing. And also the (proposition) 'if it is night,
it is day' is according to Philo true in case it is day, because,
while it begins with the false (proposition) 'it is night', it ends
with the true (proposition) 'it is day'. But according to Dio-
dorus it is false because, while it can begin - when night is
come - with the true (proposition) 'it is night', it can end with
the false (proposition) 'it is day'.
So we can fix Diodorean implication by the following definition :
20.081 If p, then q, if and only if, for every time I it is
not the case that p is true at t and q is false at t.
3. 'Connexive'' implication
20.09 (According to Diodorus) this (proposition) is true:
'if there are no atomic elements of things, then there are
atomic elements of things' . . . but those who introduce con-
nection ((TuvapTTjcjLv) say that the connected (proposition) is
sound when the contradictory (avTt,x£i[i.s:vov) of its consequent
is incompatible ((xdcx^Tai) with its antecedent. So according to
them the aforesaid connected (propositions) (20.07) are bad
({AoxOvjpa), but the following is true (<xXyfi£q) : 'if it is day, it is
day'.
20.10 A connected (proposition) is true in which the opposite
of the consequent is incompatible with the antecedent, e.g.
'if it is day, it is light'. This is true, since 'it is not light', the
opposite of the consequent, is incompatible with 'it is day'.
A connected (proposition) is false in which the opposite of the
consequent is not incompatible with the antecedent, e.g. 'if it
is day, Dion walks about'; for 'Dion is not walking about' is
not incompatible with 'it is day'.
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MEGARIAN-STOIC LOGIC
This definition is often ascribed to Chrysippus (20.11), but that
it originated with him may be doubted (20.12). It is not clear how-
it is to be understood. Perhaps we have here an ancient form of
strict implication (49.04 31.13).
4. ' Inclusive' implication
20.13 Those who judge (implication) by what is implicit
(eu-cpaaei xptvovT£<;) say that the connected (proposition) is
true when its consequent is potentially (Suvdqiei) contained in
the antecedent. According to them the (proposition) 'if it is
day, it is day' and every repetitive connected (proposition) is
probably false, since nothing can be contained in itself.
This definition too is not now fully intelligible. It seems to concern
a relation of subordination something like that which holds between
a statement about all elements of a class and one about the elements
of one of its sub-classes. No further reference to this definition is to
be found in our sources; perhaps it was only adopted by isolated
logicians of the school.
C. DISJUNCTION
We know much less about disjunction than about implication.
Apparently it formed the subject of the same sort of dispute that
there was about the definition of implication. But our texts are few
and obscure. It is only certain that two kinds of disjunction were
recognized : the complete (exclusive) and the incomplete (not exclu-
sive), of which the first is well exemplified.
1. Complete disjunction
20.14 The disjunctive (proposition) consists of (contra-
dictorily) opposed (propositions), e.g. of those to the effect
that there are proofs and that there are not proofs. . . . For as
every disjunctive is true if (and only if) it contains a true
(proposition) and since one of (two contradictorily) opposed
(propositions) is evidently always true, it must certainly be
said that the (proposition) so formed is true.
20.15 There is also another (proposition) which the Greeks
call Sis^euyuivov a£icou.a and we call disjunctum. This is of the
kind : 'pleasure is either good or bad, or neither good nor bad'.
Now all (propositions) which are disjoined (disjuncta) (within
one such proposition) are mutually incompatible, and their
opposites, which the Greeks call avn.xsiu.sva must also be
mutually opposed (contraria). Of all (propositions) which are
disjoined, one will be true, the others false. But when none of
119
THE GREEK VARIETY OF LOGIC
them at all are true, or all, or more than one are true, or when
the disjoined (propositions) are not incompatible, or when
their opposites are not mutually opposed, then the disjunctive
(proposition) will be false. They call it TOxpocSis^euyfiivov.
20.16 The true disjunctive (proposition) tells us that one of
its propositions is true, the other or others false and incompat-
ible.
These texts offer a difficulty, in the supposition that a statement
can be contradictory to more than one other. However, the practice
of the school concerning the disjunction here defined is clear: in the
sense envisaged 'p or q' is understood as the negation of equivalence
(vide infra 22.07), i.e. in such a way that just one of the two argu-
ments is true and just one false.
2. Incomplete disjunction
The surviving information about this is very vague. The best is
given by Galen, but raises the question how much of it is Megarian-
Stoic doctrine and how much Galen's own speculation:
20.17 This state of things exhibits a complete incompat-
ibility (tiXeiav paxyjv), the other an incomplete (sXXottjv)
according to which we say for example: 'if Dion is at Athens,
Dion is not at the Isthmus'. For this is characteristic of in-
compatibility, that incompatibles cannot both be the case;
but they differ in that according to the one the incompatibles
can neither both be true nor both false, but according to the
other this last may occur. If then only inability to be true
together characterizes them, the incompatibility is incomplete,
but if also inability to be false together, it is complete.
20.18 There is no reason why we should not call the propos-
ition involving complete incompatibility 'disjunctive' and that
involving incomplete incompatibility 'quasi-disjunctive'. . . .
But in some propositions not only one, but more or all
components can be true, and one must be. Some call such
'sub-disjunctive' (7cap<x8is£euY(Aeva) ; these contain only one
true (proposition) among those disjoined, independently of
whether they are composed of two or more simple propositions.
Evidently this is a matter of two different kinds of disjunctive
propositions, and so of disjunction. The first is called 'quasi-disjunc-
tion' and seems to be equivalent to the denial of conjunction:
20.181 p or q if and only if, not: p and q.
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MEGARIAN-STOIC LOGIC
Then the intended functor would be that of Shelter (43.43).
The second kind is called 'sub-disjunctive' and could be defined
by the following equivalence:
20.182 p or (also) q if and only if: if not p, then q.
This is the modern functor of the logical sum (cf. 14.1011. .
Neither of these two functors was used by the Stoics in practice,
at least so far as we can ascertain from the extant sources.
D. CONJUNCTION
20.19 What the Greeks call (jufJwue7cXeYjjtivov we call conjuncium
or copulatum. It is as follows : 'Publius Scipio, son of Paulus, was twice
consul and had a triumph and was censor and was colleague of
Lucius Mummius in the censorship.' In every conjunctive the whole
is said to be false if one (component) is false, even if the others are
true. For if I were to add to all that I have truly said about that
Scipio: 'and overcame Hannibal in Africa', which is false, then the
whole conjunctive which includes that would be false: because
that is a false addition, and the whole is stated together.
E. EQUIVALENCE
20.20 Syllogisms which have hypothetical premisses are
formed by transition from one thing to another, because of
consequence (axoXou0La) or incompatibility, each of which
may be either complete or incomplete.
20.21 The (exclusive) disjunctive premiss (Siaiperwdj) is
equivalent to the following: 'if it is not day, it is night'.
This last cited text, in which quite certainly complete disjunction
is intended (20.14 ff.) can only be understood as referring to 'com-
plete consequence' (20.20) - and then we have equivalence. In this
case we have the following definition, in which 'or' is to be under-
stood in the exclusive sense:
20.211 q completely follows from p if and only if, not : p or q.
We owe the discovery of these facts to Stakelum (20.22). Boethius,
probably drawing on a Stoic source, understands 'if A - B' in just
this sense 20.23). So it can be taken as likely that the functor of
equivalence was known to the Stoics as 'complete consequence'.
F. OTHER FUNCTORS
We also have definitions of the inferential proposition (cf. 19.15 .
This consists of a combination of conjunction with Diodorean
(certainly not Philonian) implication. Other kinds of compound
121
THE GREEK VARIETY OF LOGIC
propositions are the causal and the relative; their functors are not
definable by truth-matrices. Possibly there are further functors of
similar nature.
§21. ARGUMENTS
AND SCHEMES OF INFERENCE
A. CONCLUSIVE, TRUE, AND DEMONSTRATIVE ARGUMENTS
21.01 An argument (Xoyo^) is a system of premisses and
conclusion. Premisses are propositions agreed upon for the
proof of the conclusion, the conclusion is the proposition
proved from the premisses. E.g. in the following (argument):
'if it is day, it is light; it is day; therefore it is light', 'it is
light' is the conclusion, the other propositions are premisses.
21.02 Some arguments are conclusive (cjuvocxtixol), others
not conclusive. They are conclusive when a connected
proposition, beginning with the conjunction of the premisses
of the argument and ending with the conclusion, is true.
E.g. the argument mentioned above is conclusive, since from
the conjunction of its premisses 'if it is day, it is light' and 'it is
day' there follows 'it is light' in this connected proposition:
'if: it is day and if it is day, it is light: then it is light.'* Not
conclusive are arguments not so constructed.
This is a very important text, showing how accurately the Stoic
distinguished between a conditional proposition and implication
on the one hand, and an argument or inferential scheme and the
consequence-relation on the other. For an argument is conclusive
(cruvaxTLxo^) when the corresponding conditional proposition is true
The Stoics had a set terminology for the components of an argu-
ment. In the simplest case it has two premisses, X^jJiaTa (in the
wider sense); the first is also called XyjfjLfxa (the narrower sense), in
contrast to the second which is called izpoa'krityic, 21.04); when the
first premiss is connected, it is also called Tpomxov (21.05).
21.06 Of arguments, some are not conclusive (aTuepavToi),
others conclusive (nzpoLvrixoi) . Not conclusive are those in
which the contradictory opposite of the conclusion is not
Reading etnep el rjjjipa sax£, xal ■yjjxepa eari, cpcot; ecrdv. This reading was
called a 'monstrosity' by Heintz, whereas it is evidently the only correct one
(21.03).
122
MEGA R IAN-STOIC LOGIC
incompatible with the conjunction of the premisses, e.g. such
as: 'if it is day, it is light; it is day; therefore Dion walks
about'.
It seems to follow that the conditional sentence corresponding
to an argument must contain the functor of connexive implication
for an argument to be conclusive (cf. 20.09 f. and the commentary).
21.07 Of conclusive arguments some are true, others false.
They are true when besides the connected proposition, which
consists of the conjunction of the premisses and the conclusion,
being true, the conjunction of the premisses is also* true, i.e.
that which forms the antecedent in the connected proposition.
Again a text of the utmost importance, expressing a clear distinc-
tion between formal validity and truth. This distinction was admit-
tedly known to Aristotle (10.05 f.), but this is the first explicit
accurate formulation.
21.09 Of true arguments some are demonstrative (<x7co&eix-
tlxol) others not demonstrative. Demonstrative are those con-
cluding to the not evident from the evident, not demonstrative
are those not of that kind. E.g. the argument: 'if it is day, it is
light; it is day; therefore it is light' is not demonstrative, for
that it is light (which is evident) is its conclusion. On the other
hand, this is demonstrative: 'if the sweat flows through the
surface, there are intelligible (voyjtol) pores; the sweat flows
through the surface; therefore there are intelligible pores', for
it has a non-evident conclusion, viz. 'therefore there are
intelligible pores'.
'Intelligible' here means 'only to be known by the mind'; the
pores are not visible.
The division of arguments comprised in this last series of texts is
logically irrelevant, but of great methodological interest. It can be
presented thus:
[
demonstrative
[ true I
( conclusive I {
not demonstrative
arguments I [ not true
( not conclusive
* Omitting with Mates (21.08): xal to G\j[iniptxo\±ot..
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THE GREEK VARIETY OF LOGIC
B. NON-SYLLOGISTIC ARGUMENTS
A further interesting division shows how accurately the Stoics
distinguished between language and meta-language :
21.10 Of conclusive arguments, some are called by the name
of the genus, 'conclusive' (rapavTLxoL), others are called 'syl-
logistic'. Syllogistic are those which are either indemonstrable
(ava7r6<teixTOL) or are reduced to the indemonstrable by means
of one or more rules (tc5v 0£u.dcTcov), e.g. 'if Dion walks about,
Dion is in motion; Dion walks about; therefore Dion is in
motion'. Conclusive in the specific sense are those which do not
conclude syllogistically, those of e.g. the following kind : 'it is
false that it is day and it is night; it is day; therefore it is not
night'. Non-syllogistic, on the other hand, are arguments
which appear to resemble syllogistic ones, but do not conclude,
e.g. 'if Dion is a horse, Dion is an animal; Dion is not a horse;
therefore Dion is not an animal'.
21.11 . . . but the moderns, who follow the linguistic
expression, not what it stands for, . . . say that if the expres-
sion is formulated thus: 'if A, then B; A; therefore B,' the
argument is syllogistic, but 'B follows on A; A; therefore B'
is not syllogistic, though it is conclusive.
21.12 . . . The kind of argument which is called 'unmetho-
dically concluding' (au^OoSox; 7cepaivovTes) is e.g. this: 'it is
day; but you say that it is day; therefore you say true'.
21.13 (Those which the moderns call 'unmethodically
concluding' . . .) are such as the following: 'Dion says that it is
day; Dion says true; therefore it is day'.
21.14 . . . like the unmethodically concluding arguments
among the Stoics. When e.g. someone says: 'the first (is)
greater than the second, the second than the third, therefore
the first (is) greater than the third'.
C. FURTHER KINDS OF ARGUMENT
21.15 Those arguments too which they call 'duplicated'
(SioccpopoVevoi) are n°t syllogistic, e.g. this: 'if it is day, it is
day; therefore it is day'.
21.16 The argument): 'if it is day, it is light; it is day;
therefore it is day', and in general those which the moderns
call 'not diversely concluding' (aSt^opo^ rcepaivovTe^) . . . .
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M E G A R I A N - S T O I C LOGIC
21.17 Antipater, one of the most celebrated men of the
Stoic sect, used to say that arguments with a single premiss
can also be formed (|jiovoXy)(IU,<xtol).
21.18 From one premiss there results no (conclusive) com-
bination (colledio), though the consequence (conclusio) 'you
see, therefore you live' seemed complete to Antipater the
Stoic, against the doctrine of all (others) - for it is complete
(only) in the following way; 'if you see, then you live; you see;
therefore you live'.
21.19 Such an argument as that which says: 'it is day; not:
it is not day; therefore it is light' has potentially a single
premiss.
Further, apparently numberless, divisions of arguments are
obscure in our sources. Diogenes speaks of 'possible, impossible,
necessary and not necessary' arguments (21.20). Sextus has a
division into demonstrable and indemonstrable arguments, the
last-named being either simple or compound, and the compound
being reducible to the simple (which makes them demonstrable),
(21.21). The whole account is so vague that we are not in a position
to grasp the meaning of this division. But in Diogenes we find a
consistent doctrine of these same 'indemonstrable' arguments;
they are simply the axioms of the Stoic propositional logic, and we
consider them in the next chapter.
D. SCHEMES OF INFERENCE
The Stoics made clear distinction between a logical rule and an
instance of it, i.e. between the moods (Tpo7roi) of an argument
and the argument itself - a distinction which Aristotle applied in
practice, but without a theoretic knowledge of it.
21.22 These are some of the arguments. But their moods or
schemata (ox^octoc) in which they are formed are as follows:
of the first indemonstrable: 'if the first, then (the) second; the
first; therefore the second'; of the second: 'if the first, (then)
the second; not the second; therefore not the first'; of the
third : 'not : the first and the second ; the first; therefore not the
second'.
We have similar schemata for other arguments as well (21.23 .
even for some of the not indemonstrable (cf. 22.17). It is striking
that only numerical words occur in them as variables. One might
conjecture that this was so in Aristotle too, since in Greek the
letters of the alphabet could function as numerals; but the fact
125
THE GREEK VARIETY OF LOGIC
that Aristotle did not only use the early letters of the alphabet,
but often IT, P, and 2 as well, seems to exclude this.
Along with these homogeneous formulae the Stoics also had
'mixed' half-arguments, half-schemata. They were called 'argument-
schemata' (Xoy6Tpo7iot,).
21.24 An argument-schema consists of both, e.g. 'if Plato
lives, Plato breathes; the first; therefore the second'. The
argument-schema was introduced in order not to have to
have a long sub-premiss in long formulae, so as to gain the
conclusion, but as short as possible: 'the first; therefore the
second'.
Another example is this:
21.25 If the sweat flows through the surface, there are
intelligible pores; the first; therefore the second.
§22. AXIOM ATI ZAT I ON. COMPOUND ARGUMENTS
The Stoic propositional logic seems to have been thoroughly
axiomatized, distinction even being made between axioms and
rules of inference.
A. THE INDEMONSTRABLES
The tradition is obscure about the axioms (22.01 ; et vid. supra
21.21). We here give the definition of the indemonstrables according
to Diogenes, their description, with examples, from Sextus.
22.02 There are also some indemonstrables (av<x7c68eixToi)
which need no demonstration, by means of which every (other)
argument is woven; they are five in number according to
Chrysippus, though other according to others. They are
assumed in conclusives, syllogisms and hypotheticals (Tp07ux£>v)
22.03 The indemonstrables are those of which the Stoics
say that they need no proof to be maintained. . . . They
envisage many indemonstrables, but especially five, from
which it seems all others can be deduced.
This is no less than an assertion of the completeness of the system :
whether it is correct we cannot tell, since we do not know the
metatheorems and have only a few of the derivative arguments.
126
MEGARIAN-STOIG LOGIC
22.04 The first (indemonstrable) from a connected (pro-
position) and its antecedent yields its consequent, e.g. 'if
it is day, it is light; it is day; therefore it is light';
22.05 the second from a connected (proposition) and the
contradictory opposite (avTixst^evou) of its consequent yields
the contradictory opposite of its antecedent, e.g. 'if it is
day, it is light; it is not light; therefore it is not day' ;
22.06 the third from the negation (obccxpaTixoO) of a con-
junction together with one of its components, yields the
contradictory opposite of the other, e.g. 'not: it is day and
it is night; it is day; therefore it is not night' ;
22.07 the fourth, from a (complete) disjunctive (proposi-
tion) together with one of the (propositions) disjoined (e7te£evy-
(jtivoov) in it, yields the contradictory opposite of the other, e.g.
'either it is day or it is night; it is day; therefore it is not
night' ;
22.08 the fifth from a (complete) disjunctive (proposition)
together with the contradictory opposite of one of the dis-
joined (propositions) yields the other, e.g. 'either it is day or it
is night; it is not night; therefore it is day'.
Other less reliable sources speak of two further indemonstrables,
the sixth and seventh (22.09).
B. METATHEOREMS
The reduction of demonstrable arguments to indemonstrable, was
effected in Stoic logic by means of certain metalogical rules. One
name for such was Os(jia, but it seems that the expression OscopTjpia
was also used (22.10). We shall call them 'metatheorems', in accord-
ance with modern usage. A text of Galen shows that there were
at least four of them (22.11), but only the first and third are stated
explicitly.
22.12 There is also another proof common to all syllogisms,
even the indemonstrable, called '(reduction) to the impossible'
and by the Stoics termed 'first metatheorem' (constitutio) or
'first exposition' (expos Hum). It is formulated thus: 'If some
third is deduced from two, one of the two together with the
opposite of the conclusion yields the opposite of the other.'
This is the rule for reduction to the impossible (16.33), already
stated by Aristotle in another form.
22.13 The essentials of the so-called third metatheorem
(SepuxTCK;) look like this : if some third is deduced from two and
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THE GREEK VARIETY OF LOGIC
one (of the two) can be deduced syllogistically from others,
the third is yielded by the rest and those others.
This metatheorem is what in fact underlies the aristotelian
'direct reduction' of syllogisms, and can be formulated:
22.131 If r follows from p and q, and p from s, then r
follows from q and s (cf. 14.141).
The following is given by Alexander as the 'synthetic theorem'
(cjuvOstixov 0s:copY][jLa) :
22.14 If some (third) is deduced from some (premisses), and
if the deduced (third) together with one or more (fourth)
yields some (fifth), then this (fifth) is deduced also from those
(premisses) from which this (third) is deduced.
The rule being stated is this:
22.141 If r follows from p and g, and t from r and s,
then i follows from p, q and s;
or, if one represents the premisses with a single variable :
22.142 If q follows from p, and s from q and r, then s
follows also from p and r.
Sextus cites a similar but seemingly different metatheorem:
22.15 It should be known that the following dialectical
theorem (OscopY^oc) has been handed down for the analysis of
syllogisms : 'if we have premisses to yield a conclusion, then we
have this conclusion too potentially among these (premisses),
even if it is not explicitly (xoct* excpopav) stated.
We have two detailed examples of the apllication of this meta-
theorem, which belong to the highest development of Stoic logic.
C. DERIVATION OF COMPOUND ARGUMENTS
22.16 Of the not-simple (arguments) some consist of
homogeneous, others of not homogeneous (arguments). Of
not homogeneous, those which are compounded of two first
indemonstrables (22.04), or of two second (22.05). Of not
homogeneous, those which (are compounded) of second and
third* (22.06), or in general of such (dissimilars). An example of
* Reading xal Tptxou in the lacuna with Kochalsky.
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MEGARIAN-STOIC LOGIC
those consisting of homogeneous (arguments) is the following:
'if it is day*, then if it is day it is light; it is day; therefore it
is light'. . . . For we have here two premisses, (1) the connected
proposition: 'if it is day* *, then if it is day it is light', which
begins with the simple proposition 'it is day' and ends with
the not simple, connected proposition 'if it is day it is light';
and (2) the antecedent in this (first premiss:) 'it is day'. If
by means of the first indemonstrable we infer from those the
consequent of the connected (proposition, viz.) 'if it is day it
is light', then we have this inferred (proposition) potentially in
the argument, even though not explicitly stated. Putting this
now together with the minor premiss of the main argument,
viz. 'it is day', we infer by means of the first indemonstrable:
'it is light', which was the conclusion of the main argument.
22.17 That is what the arguments are like which are com-
pounded from homogeneous (indemonstrables). Among the
not homogeneous is that propounded by Ainesidemus about
the sign, which runs thus: 'if all phenomena appear similarly
to those who are similarly disposed, and signs are phenomena,
then signs appear similarly to all those who are similarly
disposed; signs do not appear similarly to all those who are
similarly disposed; phenomena appear similarly to all those
who are similarly disposed ; therefore signs are not phenomena'.
This argument is compounded of second and third indemon-
strables, as we can find out by analysis. This will be clearer if
we put the process in the form of the schema of inference:
'if the first and second, then the third ; not the third ; the first;
therefore not the second'. For we have here a connected (propo-
sition) in which the conjunction of the first and second forms
the antecedent, and the third the consequent, together with
the contradictory opposite of the consequent, viz. 'not the
third'. Hence we infer by means of the second indemonstrable
the contradictory opposite of the antecedent, viz. 'therefore
not: the first and the second'. But this is potentially contained
in the argument, as we have it in the premisses which yield it,
though not verbally expressed. Putting it * * * together with the
other premiss, the first, we infer the conclusion (of the main
argument), 'not the second', by means of the third indemon-
strable.
* Adding with Kochalsky: si yjfiipa saxtv.
* * Completing the text as before.
* * * reading 07rsp instead of a7rep, with Kochalsky.
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THE GREEK VARIETY OF LOGIC
D. FURTHER DERIVED ARGUMENTS
According to Cicero (22.18) the Stoics derived 'innumerable'
arguments in similar ways.
22.19 The said (Chrysippus) says that it (the dog) often*
applies the fifth indemonstrable, when on coming to the
meeting of three roads it sniffs at two down which the game
has not gone and immediately rushes down the third without
sniffing at it. The sage says in fact that it virtually infers:
the game has gone down this, or this, or that; neither this nor
this; therefore that.
22.21 If two connected (propositions) end in contradictorily
opposed (consequents) - this theorem is called (the theorem)
'from two connecteds' (Tpo7cixc5v) - the (common) antecedent
of the two is refuted .... This argument is formed according to
the schema of inference : 'if the first, the second ; if the first * *,
not the second; therefore not the first'. The Stoics give it
material expression (i.e. by a substitution) when they say
that from the (proposition) 'if you know that you are dead
(you are dead if you know that you are dead) you are not dead'
there follows this other: 'therefore you do not know that you
are dead'.
22.22 Some argue in this way: 'if there are signs, there are
signs; if there are not signs, there are signs; there are either no
signs or there are signs; therefore there are signs'.
§ 23. THE LIAR
The Stoics and above all the Megarians devoted much attention
to fallacies. Some of the ones they considered derive from the problem
of the continuum and belong to mathematics in the narrower sense
of that word ; the rest are mostly rather trifles than serious logical
problems (23.01). But one of their fallacies, 'the Liar' (^£uSo[X£vo<;)
has very considerable logical interest and has been deeply studied
by logicians for centuries, in antiquity, the middle ages, and the
20th century. The Liar is the first genuine semantic fallacy known to
us.
* Sia tcXeiovcov: this could also mean 'the (argument) from the more'; but I
follow Mates (22.20) since (1) we know of no such indemonstrable, and (2) the
argument is reducible to the simple fifth indemonstrable.
* * omitting ou with Koetschau.
130
MEGARIAN-STOIC LOGIC
A. HISTORY
In St. Paul is to be found the following notable text:
23.02 One of themselves, a spokesman of their own, has
told us : The men of Crete were ever liars, venomous creatures,
all hungry belly.
According to various sources (23.03) this spokesman was Epimeni-
des, a Greek sage living at the beginning of the 6th century b.c.
Hence the Liar is often called after him, but wrongly, for Epimenides
was clearly not worrying about a logical paradox. Plato, too, who con-
sidered similar problems in the Euthydemus (ca. 387 B.C.; 23.04; did
not know the Liar. But Aristotle has it in the Sophistic Refutations,
about 330 b.c. (23.05). Now that is just the period when Eubulides
was flourishing, to whom Diogenes Laertius explicitly ascribes the
discovery of the Liar (18.02). After that, Theophrastus wrote three
books on the subject (23.06), Chrysippus many more, perhaps
twenty-eight (23.07). How much people took the problem to heart
at that time can be seen from the fact that one logician, Philetas of
Cos (ca. 340-285 b.c), died because of it:
23.08 Traveller, I am Philetas; the argument called the
Liar and deep cogitations by night, brought me to death.
B. FORMULATION
In spite of this interest and the extensive literature about the
Liar, we no longer possess Eubulides's formulation of the antinomy,
and the versions that have come down to us are so various that it
is impossible to determine whether a single formula underlies them
all, and which of the surviving ones has been considered by com-
petent logicians. Here we can only give a simple list of the most
important, collected by A. Riistow (23.09). They seem to fall into
four groups.
I.
23.10 If you say that you lie, and in this say true, do you
lie or speak the truth ?
23.11 If I lie and say that I lie, do I lie or speak the truth?
II
23.12 If you say that you lie, and say true, you lie; but
you say that you lie, and you speak the truth; therefore you
lie.
23.13 If you lie and in that say true, you lie.
131
THE GREEK VARIETY OF LOGIC
III.
23.14 I say that I lie, and (in so saying) lie; therefore I
speak the truth.
23.15 Lying, I utter the true speech, that I lie.
IV.
23.16 If it is true, it is false ; if it is false, it is true.
23.17 Whoso says 'I lie', lies and speaks the truth at the
same time.
The relation of the four groups to one another is as follows : the
texts of the first group simply posit the question: is the Liar true or
false ? Those of the second conclude that it is true, of the third that
it is false. The texts of the fourth group draw both conclusions
together; the proposition is both true and false.
C. EFFORTS AT SOLUTION
Aristotle deals with the Liar summarily in that part of his
Sophistic Refutations in which he discusses fallacies dependent on
what is said 'absolutely and in a particular respect':
23.18 The argument is similar, also, as regards the problem
whether the same man can at the same time say what is both
false and true: but it appears to be a troublesome question
because it is not easy to see in which of the two connections
the word 'absolutely' is to be rendered - with 'true' or with
'false'. There is, however, nothing to prevent it from being
false absolutely, though true in some particular respect (nfi) or
relation (tiv6<;), i.e. being true in some things though not
true absolutely.
It has been said (23.19) that the difficulty is here 'quite unresolved,
and indeed unnoticed', and indeed Aristotle has not solved our
antinomy nor understood its import. Yet, as is so often the case
with this pastmaster, he reveals a penetrating insight into the
principle of the medieval and modern solutions - the necessity of
distinguishing different aspects, levels as we now say, in the Liar.
Worth noting too, is Aristotle's standpoint of firm conviction that
a solution is discoverable. This conviction has remained the motive
power of logic in this difficult field.
The solution of Chrysippus has reached us in a very fragmentary
papyrus, written moreover in difficult language. Its essential, and
most legible, part is as follows* :
* Thanks are due to Prof O. Gigon for help with this text.
132
MEGARIAN-STOIC LOGIC
23.20 The (fallacy) about the truth-speaker and similar
ones are to be . . . (solved in a similar way). One should not say
that they say true and (also) false; nor should one conjecture
in another way, that the same (statement) is expressive of true
and false simultaneously, but that they have no meaning at all.
And he rejects the afore-mentioned proposition and also the
proposition that one can say true and false simultaneously and
that in all such (matters) the sentence is sometimes simple,
sometimes expressive of more.
The most important words in this text are cn)[i.ar.vofiivoi> ziXzac,
(X7ro7rAavc5vTai, translated '(that) they have no meaning at all'.
The Greek phrase is ambiguous as between (1) that whoever states
the Liar attributes a false assertion to the proposition, and (2) that
he says something which has no meaning at all. The fragmentary
context seems to indicate the second interpretation as the correct
one, but it is impossible to be certain of this. If it is correct, Ghrysip-
pus's solution is that the Liar is no proposition but a senseless
utterance, which would be a view of the highest importance. The
Aristotelian attempt to solve it is definitely rejected in this text.
133
IV. THE CLOSE OF ANTIQUITY
§24. PERIOD OF COMMENTARIES
AND HANDBOOKS'
A. CHARACTERISTICS AND HISTORICAL SURVEY
With the end of the old Stoa there begins a period into which
hardly any research has been done. However, on the basis of the
few details known to us we may suppose with great probability that
the formal logic of this period was of the following kind :
1. The period is not a creative one. No new problems or original
methods such as those developed by Aristotle and the Megarian-
Stoic school are to be found.
2. Yet, apparently right up to the fall of the Roman empire,
individual scholarly works appeared. Some earlier methods were
improved, the material was systematized and sometimes developed.
There were even not wanting genuinely gifted logicians, among the
best of whom was Alexander of Aphrodisias.
3. The logical literature consisted chiefly of two kinds of work:
big commentaries, mainly on Aristotle, and handbooks.
4. As to their content, we discern mostly a syncretizing tendency
in the sense that Aristotelian and Stoic-Megarian elements are
mingled, Stoic methods and formulations being applied to Aristo-
telian ideas.
Lack of monographs makes it impossible to survey the state of the
logical problematic during the period, and we limit ourselves to the
choice of some particular doctrines so far found in the mass of
commentaries and handbooks. But first some of the most important
thinkers must be named.
The first well-known logicians of this period are Galen and the
less notable Apuleius of Madaura whose handbooks have survived;
the former is the subject of the only monograph on the period
(24.01). In the 3rd century a.d. we find Alexander of Aphrodisias,
already mentioned, one of the best commentators on the whole
Aristotelian logic, and unlike Galen and Apuleius a fairly pure
Aristotelian. Porphyry of Tyre lived about the same time, and
composed an Introduction (ziGOLycxtyri) to the Aristotelian categories.
In it he systematized the doctrine of the predicables (11.06ff.),
giving a five-fold enumeration: genus, specific difference, species,
property and accident (24.02). This work was to be basic in the
Middle Ages. Later logicians include Iamblichus of Chalcis, not to
be taken very seriously, Themistius (both these in the 4th century
a.d.), Ammonius Hermeae (5th century), Martianus Capella, author
134
TREE OF PORPHYRY
of a handbook which formed an important link between ancient
and later logic (5th century), AmmoniuB the Peripatetic, Simplicius
(6th century), who was another of the better commentators on
Aristotle, and finally Philoponus (7th century), but these have
little importance so far as we can judge. On the other hand the last
Roman logician, Boethius (ca. 480-524) is of fairly considerable
importance both because his works became a prime source for the
Scholastics and also because he transmits doctrines and methods
not mentioned elsewhere, though he himself was only a moderate
logician. With his execution the West enters on a long period without
any logic worth speaking of.
B. THE TREE OF PORPHYRY
Of the commentators' discoveries the 'tree of Porphyry' has
certainly achieved the greatest fame. While it can be regarded as
only a compendium of Aristotelian doctrines it has great importance
as comprising (1) a system of classification, which was not to the
fore in Aristotle's thought (11.13), and (2) an extensional view of
terms. First we give the text:
24.03 Let what is said in one category now be explained.
Substance (ouctioc) is itself a genus, under this is body, and under
body is living (s^uxov) body, under which is animal. Under
animal is rational (Xoyixov) animal, under which is man. Under
man are Socrates and Plato and individual (xorra uipo<;) men.
But of these, substance is the most generic and that which is
genus alone ; man is the most specific and that which is species
alone. Body is a species of substance, a genus of living body.
The following text shows how thoroughly extensional a view is
being taken:
24.04 (Genus and species) differ in that genus contains
(■7T£ptix£L) its species, the species are contained in but do not
contain their genus. For the genus is predicated of more things
than the species.
This conception is carried so far that one can here properly speak
of a beginning of calculus of classes. At the same time Porphyry
makes a distinction which corresponds fairly closely to the modern
distinction between extension and intension (36.10, 45.03) - or.
again, between simple and personal supposition (27.15). For among
a number of definitions of the predicables, he has :
135
CLOSE OF ANTIQUITY
24.05 The philosophers . . . define, saying that genus is what
is predicated essentially (ev tw tl eoti) of several things
differing in species.
24.06 The genus differs from the difference and the common
accidents in that, while the difference and the common acci-
dents are predicated of several things differing in species, they
are not predicated essentially but as qualifying (ev tco o7rot6v tl
e<mv). For when we ask what it is of which these are predicated,
we answer with the genus; but we do not answer with the
differences or accidents. For they are not predicated essentially
of the subject but rather as qualifying it. For on being asked
of what quality man is, we say that he is rational, and to the
question of what quality crow is, we answer that it is black.
But rational is a difference, and black an accident. But when
we are asked what man is, we answer that he is animal, animal
being a genus of man.
C. EXTENSION OF LOGICAL TECHNIQUE
Among the most important achievements of this period are two
devices which so far as we know were unknown to Aristotle and the
Stoics, viz. (1) identification of variables, (2) substitution of sen-
tential forms for variables.
1. Alexander of Aphrodisias
The first is to be found in Alexander in a new proof of the con-
vertibility of universal negative sentences :
24.07 If someone were to say that the universal negative
(premiss) does not convert, (suppose) A belongs to no B; if
(this premiss) does not convert, B belongs to some A; there
results in the first figure (the conclusion that) to some A A
does not belong, which is absurd.
Alexander here makes use of the fourth syllogism of the first
figure (Ferio: (13.06), which in Aristotle's presentation runs: 'if A
belongs to no B, but B to some C, then to some C A cannot belong'.
He identifies C with A - i.e. substitutes one variable for the other,
and obtains: 'if A belongs to no B, but B to some A, then to some A
A does not belong'. That is the novelty of the process.
This is consonant with Alexander's clear insight into the nature
of laws of formal logic. He seems to have been the first to make
explicit the distinction between form and matter, and at the same
time to have come close to an explicit determination of the notion
of a variable.
136
IMPLICATION
24.08 He (Aristotle) introduces the use of letters in order
to show us that the conclusions are not produced in virtue
of the matter but in virtue of such and such a form
(oXWol) and composition and the mood of the premisses; the
syllogism concludes . . . not because of the matter, but because
the formula (au^uyta) is as it is. The letters show that the
conclusion is of such a kind universally and always and for
every choice (of material).
2. Boethius
A further development of the technique of formal logic is to be
found in Boethius. He is evidently aiming at the formulation of a
rule of substitution for propositional variables ; this is not given in the
form of such a rule, but in a description of the structure of formulae.
Again we have a fairly clear distinction between form and matter in
a proposition, a distinction which was to play a great part in later
history :
24.09 We shall now show the likenesses and differences
between simple propositions and compound hypothetical ones.
For when the (hypothetical propositions) which consist of
simple ones are compared with those which are compounded of
two hypotheticals, (one sees that) the sequence (in both cases)
is the same and the relation (of the parts to one another)
remains, only the terms are doubled. Since the places which are
occupied by simple propositions in those hypotheticals consist-
ing of simple propositions, are occupied in hypotheticals consist-
ing of hypotheticals by those conditions in virtue of which those
(component) propositions are said to be joined and linked
together. For in the proposition which says: 'if A is, B is',
and in that which says: 'if, if A is, B is, (then) if C is, D is'
the place occupied in that consisting of two simple propositions
by that which is first: 'if A is', in the proposition consisting of
two hypotheticals is occupied by that which (there) is first: 'if,
if A is, B is'.
If we remember the Stoic distinction between argument and mood
(21.22) the last two texts do not seem very original; but they are the
first in which an explicit statement of the distinction is found.
D. FRESH DIVISION OF IMPLICATION
It is Boethius again who gives a fresh division of implication :
24.10 Every hypothetical proposition is formed either by
connection (connexionem) ... or by disjunction. . . . But since
137
CLOSE OF ANTIQUITY
it has been said that the same thing is signified by the connec-
tives (conjunctione) 'si' and 'cum' when they are put in hypo-
thetical sentences, conditionals can be formed in two ways:
accidentally, or so as to have some natural consequence.
Accidentally in this way, as when we say: 'when fire is hot,
the sky is round'. For the sky is round not because fire is hot,
but the sentence means that at what time fire is hot, the
sky is round. But there are others which have within them
a natural consequence, . . . e.g. we might say: 'when man is,
animal is'.
There is here, as often elsewhere, a certain obscurity in Boethius's
thought (24.11). Apart from that, his division of implication is
something of a backward step in comparison with the Stoic dis-
cussions of the subject (vide supra 20.05 ff.). Yet the text just cited is
important for our history, being an evident starting-point for scholas-
tic speculations about implication.
Hence also we mention the following details of Boethius's doc-
trine about propositional functors. He often seems to use lsV (24.12)
as a symbol of equivalence (cf. 20.20 ff.). The sense of the expression
kauV is ambiguous. On the one hand, we find - and for the first
time - a definition in the sense of non-exclusive alternation (logical
sum : cf. 20.17, 30.18, 40.11, 41.18) :
24.13 The disjunctive proposition which says (proponit) :
'either A is not or B is not' is true (fit) of those things which
can in no way co-exist, since it is also not necessary that
either one of them should exist; it is equivalent to that com-
pound proposition in which it is said : 'if A is, B is not'. ... In
this proposition only two combinations yield (valid) syllogisms.
For, if A is, B will not be, and if B is, A will not be. . . . For
if it is said: 'either A is not or B is not', it is said: 'if A is, B
will not be', and 'if B is, A will not be'.
First we have here Sheffer's functor ('not p or not q'-, 43.43);
secondly this text contains an exact definition of the logical sum.
The essential idea can be formulated :
24.131 Not p or not q if and only if : if p then not q.
Putting therein 'not-p' for 'p' and 'not-g' for lq\ we get by the
principle of double negation:
24.132 p or q if and only if: if not-p then q.
138
HYPOTHETICAL SYLLOGISMS
On the other hand, Boethius defines in analogous fashion his
lsV in the sense of equivalence by means of the same 'auV - which
therefore and in this case has the sense of aegaied equivalence
(p or q but not both, and necessarily one of the two) (24.14).
It is also worth remarking that Boethius regularly uses the
principle of double negation and a law analogous to 24.21.
E, BOETHIUS'S HYPOTHETICAL SYLLOGISMS
We here give the list of Boethius's hypothetical syllogisms. They
seem to be the final result of Stoic logic, if understood as laws of the
logic of propositions. Our supposition that Boethius aspired to a rule
of substitution for propositional variables (cf. 24.09), requires them to
be so understood. They would be the final result of Stoic logic in the
sense that they are practically the only part of this logic that was
preserved by Boethius for the Middle Ages.
24.15 If A is, B is; but A is; therefore B is.
24.16 If A is, B is; but B is not; therefore A is not.
24.17 If A is, B is, and if B is, C must be; but then: if A is,
C must be.
24.18 If A is, B is, and if B is, C too must be; but C is not;
therefore A is not.
24.19 If A is, B is ; but if A is not, C is ; I say therefore that
if B is not, C is.
24.20 If A is, B is not; if A is not, C is not; I say therefore
that if B is, C is not.
24.21 If B is, A is; if C is not, A is not; on this supposition
I say that if B is, it is necessary that C is not.
24.22 If B is, A is; if C is not, A is not; I say therefore: if
B is is, C will be.
24.23 If one says : 'either A is or B is', (then) if A is, B \x\\\
not be; and if A is not, B will be; and if B is not, A will be;
and if B* is, A will not be.
24.25 The (proposition) that says: 'either A is not or B is
not', certainly means this, that if A is, B cannot be.
Boethius developes these syllogisms by substituting a conditional
proposition for one or both variables (cf. 24.09) ; in so doing he treats
the negation of a conditional as the conjunction of the antecedent
with the negation of the consequent, according to the law. which is
not expressly formulated:
24.251 Not: if p, then q, if and only if: p and not-g.
* omitting non with van den Driessche (24.24).
139
CLOSE OF ANTIQUITY
Finally he applies the law of double negation (cf. 20.041), thus
gaining eighteen more syllogisms (24.26).
F. ALTERATIONS AND DEVELOPMENT OF THE CATEGORICAL
SYLLOGISTIC
24.27 But Ariston the Alexandrian and some of the later
Peripatetics further introduce five more moods (formed from
those) with a universal conclusion: three in the first figure,
two in the second figure, which yield particular conclusions.
(But) it is extremely foolish to conclude to less from that to
which more is due.
This text is not very clear. But its difficulty is somewhat lessened
if we suppose that a combination of two Aristotelian rules is envi-
saged: (1) that allowing a universal conclusion to be weakened to
the corresponding particular (13.23), (2) that yielding a further
conclusion by conversion of the one first obtained. Then the follow-
ing would be the moods intended :
24.271 A to all B; B to all C; therefore A to some C
(Barbari).
24.272 A to no B; B to all C; therefore to some C, A not
(Celaront).
24.273 A to all B; B to some C ; therefore C to some A
(Dabiiis).
ZZk.Zlb B to no A; B to all C; therefore to some C, A not
(Cesaro).
24.275 B to all A; B to no C ; therefore to some C, A not
(Camestrop).
Beyond these, Galen transmits a further mood of this kind in the
third figure (24.28):
24.281 A to all B; C to all B; therefore C to some A
(Daraptis).
These formulae all have a Stoic rather than an Aristotelian form.
In fact from Apuleius on, such alteration of the old laws into rules
is more or less standard practice, especially in Boethius.
A further precision given to the Aristotelian syllogistic is in the
famous logical square. This figure is first found in Apuleius again. It
looks like this:
140
FOURTH FIGURE
24.29
Contrariae
vd
incongruae
0
c
iS
universal affirmative
W
universal negal ive
1
all pleasure is good
no pleasure is good
some pleasure is good
some pleasure is not good
particular affirmative
particular negative
Subconlrariae
vel
sub pares
G. THE SUPPOSEDLY FOURTH FIGURE
In an anonymous fragment, belonging possibly to the 6th cen-
tury, we read :
24.30 Theophrastus and Eudemus also added other formu-
lae to those of Aristotle in the first figure . . . many moderns
have thought to form the fourth figure therefrom, citing
Galen as the author of this intention.
But this allegedly 'Galenic' figure is not to be found in him. On
the contrary he plainly states that there are only three figures :
24.31 These syllogisms are called, as I have said, categorical ;
they cannot be formed in more than the three figures men-
tioned, nor in another number in each (of these figures); this
has been shown in the treatises On Demonstration.
J. Lukasiewicz was able to explain by means of another anony-
mous fragment how nevertheless the discovery of the fourth figure
could be credited to Galen (24.32). This fragment is not without
historical interest even apart from this question:
24.33 Of the categorical (syllogism) there are two kinds;
the simple and the compound. Of the simple syllogism there
are three kinds: the first, the second, and the third figure. Of
the compound syllogism there are four kinds: the first, the
second, the third, and the fourth figure. For Aristotle says
that there are only three figures, because he looks at the simple
syllogisms, consisting of three terms. Galen, however, says
in his Apodeidic that there are four figures, because he looks at
the compound syllogisms consisting of four terms, as he has
found many such syllogisms in Plato's dialogues.
141
CLOSE OF ANTIQUITY
24.34 The categorical syllogism
simple, as ( in) Aristotle compound, as (in) Galen
Figure 1,2,3
Compound figure
1 to 1, 1 to 2, 1 to 3, 2 to 2, 2 to 1, 2 to 3, 3 to 3, 3 to 1, 3 to 2.
1 to 1
1
Compound figure
syllogistic :
1 to 2 1 to 3
2 3
2 to 3
4
2 to 2 3 to 3
since no syllogism
arises from two
negatives or two
particulars.
1 to 1, as in the Alcibiades
unsyllogistic :
2 to 1 3 to 1 3 to 2
2 3 4
useful
The numerals denote the successive figures, and the author
means that a valid compound syllogism can be formed in four
different ways, viz. when of the two simple syllogisms from which it
is composed
(1) both are in the first figure,
(2) the first is in the first, the second in the second figure.
(3) the first is in the first, the second in the third,
(4) the first is in the second, the second in the third.
Those are the four figures. So there is no question of a fourth figure of
simple syllogism, which was only ascribed to Galen by a misunder-
standing. Yet the unknown scholiast (24.30), in falling a victim to
this misunderstanding at least made the principle of the fourth
figure another interpretation of the indirect moods of Theophrastus
(17.10).
142
PONS ASINORUM
H. PONS ASINORUM
Here we should introduce a scheme which was to become famous
in the Middle Ages as the pons asinorum or 'asses' bridge' 'I'l.Xil). .
It is to be found in Philoponus*, and is an elaboration of the
Aristotelian doctrine of the inuentio medii (14.29;. Although it
belongs to methodology rather than logic, it is relevant to the latter
also. The scheme seems typical of the way in which the commentators
developed the syllogistic. In Philiponus the lines are captioned in the
figure itself. For graphical reasons we put these comments after-
wards and refer to them by numbers.
24.35**
GOOD
A
What follows on the good
helpful, eligible,
to be pursued,
suitable, desirable,
profitable,
expedient.
What is alien to the good: -4
imperfect, to be
fled from, harmful,
bad, ruinous, alien,
unprofitable.
What the good
follows upon:
happiness, natural
well-being, final
cause, perfect,
virtuous life.
PLEASURE
E
What follows on
pleasure:
movement, natural
activity, unimpeded
life, object
of natural desire,
undisturbed,
eligible.
What is alien to
pleasure:
disease, labour,
fear, need,
unnatural movement.
What pleasure
follows upon:
health, good
repute, virtuous
life, plenty,
good children, freedom
from pain, comfort.
1) Unsyllogistic, because of concluding in the second figure
from two universal affirmative (premisses).
2) Universal negative (conclusion) in the first and second
(figures) by two conversions.
3) Particular affirmative (conclusion in the first and third)
figures by conversion of the conclusion.
4) Universal negative (conclusion) in the first and second
figure.
* Thanks are due to Prof. L. Minio-Paluello for pointing out this passage.
* * For typographical reason the words in the figure are set in small type,
though they belong to the quotation.
143
CLOSE OF ANTIQUITY
5) Universal affirmative (conclusion) in the first figure.
6) Unsyllogistic, from two universal negatives.
7) Particular negative (conclusion) in the third and first
(figure) through conversion of the minor (premiss).
8) Unsyllogistic, since the particular does not convert, and in
the first (figure) because (the syllogism) has a negative minor
(premiss).
9) Particular affirmative (conclusion) in the third and first
figures by conversion of the minor premiss.
I. ANTICIPATION OF THE LOGIC OF RELATIONS
Finally we shall speak of a detail which had no influence on the
later development of logic, but which yet may be reckoned an
ingenious anticipation of the logic of relations. Galen, dividing
syllogisms in his Introduction, distinguishes first between categorical
and hypothetical syllogisms, thus separating term- and class-logic;
he then adds a further class:
24.36 There is still a further, third class of syllogisms,
useful for demonstration, which I characterize as based on
relation. Aristotelians claim that they are counted as cate-
goricals. They are not a little in use among the Sceptics,
Arithmeticians and experts in calculation in certain arguments
of this kind: 'Theon possesses twice what Dion possesses; but
Philo too possesses twice what Theon possesses; therefore
Philo possesses four times what Dion possesses.'
This is in fact a substitution in a law of the logic of relations, and
it is remarkable that Galen divides his logic just as Whitehead and
Russell were to do in the 20th century. The content of his logic of
relations is of course very poor, and he thinks that such laws are
reducible to categorical syllogisms (24.37), which is a regress from
the position of Aristotle.
SUMMARY
To summarize the results of post-Aristotelian antiquity we can
say:
1. Propositional logic was then created. Some theorems of this
kind were already known to Aristotle, sometimes even stated with
propositional variables: but these were rather obiter dicta than
systematically presented. In the Stoics on the other hand we meet
systematic theory developed for its own sake.
2. This system is based on a fairly well worked out semantics,
and it was expressly stated in the Stoic school that it was concerned
144
SUMMARY
neither with words nor psychic images, hut with objective meanings,
the lecla. We have therefore to thank them for a fundamental
thesis which was to play a great part in the history of logic.
3. Megarian-Stoic logic contained an astonishingly exact analysis
of proposition-forming functors: we find correctly formed truth-tables
and a more intricate discussion of the meaning of implication than
we seem yet to have attained in the 20th century.
4. In this period the method is formalistic. Unambiguous
correlation of verbal forms to lecla being presupposed, attention is
exclusively directed to the syntactical structure of expressions.
The application of this method and the logical subtlety shown by the
Stoics must be deemed quite exemplary.
5. This formalism is accompanied by a significant extension of
logical technique, shown in the clear distinction between propositional
functions and propositions themselves, the method of identification
of variables, and the application of the rule permitting substi-
tution of propositional functions for propositional variables.
6. Propositional logic is axiomatized, and a clear distinction
drawn between laws and melalheorems.
7. Finally we have to thank the Megarian school for propound-
ing the first important logical antinomy - the Liar - which for
centuries remained one of the chief problems of formal logic, and is
so even today. So without exaggeration one can say that the achieve-
ments of this period make up antiquity's second basic contribution
to formal logic.
145
PART III
The Scholastic Variety of Logic
§25. INTRODUCTION TO SCHOLASTIC LOGIC
A. STATE OF RESEARCH
At the present time much less is known about the history of
scholastic than of ancient logic. The reason is that when Scholasti-
cism ceased to be disparaged at the end of the nineteenth century,
there was at first little revival of interest in its formal logic. This
lack of interest is shown in the fact that of more than ten thousand
titles of recent literature on Thomas Aquinas (up to 1953), very
few concern his formal logic. There are indeed earlier works treating
of questions of the literary history of scholastic logic - Grabmann
having done most to find and publish texts -, but the investigation
of their logical content only began with Lukasiewicz's paper of
1934 (25.01), pioneering in this field too. Under his influence some
notable medievalists, e.g. besides Grabmann, K. Michalski, applied
themselves to logical problems, and from his school there came the
first work, well and systematically prepared, on medieval logic, the
paper on the propositional logic of Ockham by J. Salamucha (1935)
(25.02). A number of texts and treatises followed, those of Ph. Boeh-
ner O.F.M. and E. Moody in the forefront. Today there is quite a
group at work, though as yet a small one.
But we are still at the beginning. Arabian and Jewish logic has
hardly been touched ; texts and treatises are alike lacking. In the
western domain some texts of Abelard have been published for the
12th century; for the 14th and 15th centuries we have hardly
anything, either new editions of texts or works on them; the 13th
century is almost completely inaccessible and unknown. For this
last, besides the (fairly) reliable older editions of the works of
Thomas Aquinas and (some works) of Duns Scotus, Peter of Spain
and William of Shyreswood are available only in provisional editions.
For the 14th century there is an edition of the first book of Ockham's
Summa, and one of a small work ascribed to Burleigh.*
Altogether we must say that the present state of research permits
no general survey of the sources, growth and details of scholastic
logic.
B. PROVISIONAL PERIODS
However, on the basis of the works of Ph. Boehner, E. Moody,
L. Minio-Paluello, and of the ever-growing number of general
studies of medieval philosophy, the history of medieval logic can be
provisionally divided into the following periods :
* The late Fr. Ph. Boehner was working on a critical edition of another work
of Burleigh's and of the Perutilis Logica of Albert of Saxony.
148
PERIODS
1. transitional period: up to Abelard. So far as we know this is
not remarkable for any logical novelties, and acquaintance even with
earlier achievements was very limited.
2. creative period: beginning seemingly after Abelard, about
1150, and lasting to the end of the 13th century. Former achievement-
now became known in the West, partly through the Arabs, partly (as
L. Minio-Paluello has shown*) directly from Byzantium. At the
same time work began on new problems, such as the proprieties
lerminorum, properties of terms. By about 1260 the essentials of
scholastic logic seem to have taken shape and been made widely
known in text-books. The best known book of this kind, and the
most authoritative for the whole of Scholasticism - though by no
means the first or the only one - is the Summulae Logicales of
Peter of Spain.
3. period of elaboration: beginning approximately with William
of Ockham (ob. 1349/50)** and lasting till the close of the Middle
Ages. No essentially new problems were posed, but the old were
discussed very thoroughly and very subtly, which resulted in an
extremely comprehensive logic and semiotic.
So little is known of the whole development that we are unable
even to name only the most important logicians. We can only say
with certainty that the following among others exercised great
influence :
in the 12th century: Peter Abelard (1079-1142);
in the 13th century : Albert the Great (1 193-1280) ;
Robert Kilwardby (ob. 1279),
William of Shyreswood (ob. 1249),
Peter of Spain (ob 1277);
in the 14th century: William of Ockham (ob. 1349/50),
John Buridan*** (ob. soon after 1358),
Walter Burleigh (ob. after 1343),
Albert of Saxony (1316-1390),
Ralph Strode (ca. 1370) ;
in the 15th century: Paul of Venice (ob. 1429),
Peter Tarteret (wrote between 1480 and
1490),
Stephanus de Monte
Appearance in this list comports no judgment of worth, especially
as we hardly ever know whether a logician was original or only
a copyist.
* Verbal communication from Prof. L. Minio-Paluello to whom the author is
obliged for much information about the 12th and beginning of the 13th century.
** Ockham's productive period in logic was wholly prior to 1328/29 ^25.03).
* * * Ph. Boehner states that John of Cornubia (Pseudo-Scotus) may belong to
the same period.
149
SCHOLASTIC LOGIC
C. THE PROBLEM OF SOURCES
Even the question of the literary sources for Scholasticism's new
logical problems is not yet satisfactorily answered. The works of
Aristotle provide some starting-points for the semiotic, especially
the first five chapters of the Hermeneia and the Sophistic Refutations.
Recent research shows that the latter had a decisive influence on
the scholastic range of problems*. But even the early scholastic
theory of the 'properties of terms' is so much richer and more many-
sided than the Aristotelian semiotic, that other influences must be
supposed. Grammar was certainly an important one: so far as we
can tell, that was the basis on which the main semiotic problems
were developed without much outside influence - e.g. the whole
doctrine of supposition, the growth of which can be traced with some
continuity.
We have no more certain knowledge about the origin of the
'consequences'. Boethius's teaching about hypothetical sentences
(rather than about hypothetical syllogisms) was undoubtedly very
influential.** I.Thomas's recent inquiries (cf. footnote on 30.04)
point to the Topics as a principal source; the Stoic fragments do not
seem to have been operative, at least directly, although the Outlines
of Pyrrhonism of Sextus Empiricus were already translated into
Latin in the 14th century (25.04). We do find doctrines here and
there which are recognizably Stoic, but in scholastic logical literature
as a whole Stoic logic seems to have been known only in the (obscure)
form of Boethius's syllogisms. But these do not underlie the con-
sequences, since even in fairly late works the two are treated in
distinction. Probably scholastic propositional logic is a rediscovery,
starting from hints in the Topics and perhaps also the Hermeneia,
rather than a continuation of Stoic logic.
Arabian logicians certainly exercised some influence, though
perhaps less than has commonly been supposed. But hardly any
research has been done on this subject.***
D. LOGIC AND THE SCHOOLS
The opinion has often been expressed in writings on the history
of scholastic logic that it can be divided firstly according to schools,
as it might be into nominalist and realist logic, secondly according
to faculties, and so into an 'artistic' and a 'theological' logic. But
these divisions are little relevant to formal logic as such. More
* Verbal communication from Prof. Minio-Paluello.
* * Prof. E. Moody has remarked on this to me.
*** I. Madkour's VOrganon d'Aristote dans le monde Arabe (25.05) is quite
inadequate. Prof. A. Badawi in Cairo has published and discussed a series of
Arabic logical texts but unfortunately only in Arabic. Communications received
from him indicate the presence of many interesting doctrines.
150
METHOD
modern research has shown that a number of logicians belonging
to sharply opposed philosophical schools, treated of just the same
range of problems and gave the same answers. Thus in every case
we have met there is but one doctrine of supposition, and differences
are either to be ascribed to personal idiosyncrasy rather than
philosophical presupposition, or else are more epistemological than
logical. Any contrast between artistic and theological logic is hardly
more in place. In the middle ages logic was always part of the
curriculum of the faculty of arts, but no-one was admitted to the
study of theology without having become Baccalaureun artium.
Hence the chief theological works of this period presuppose and use
the full range of 'artistic' logic. We should maintain only two
distinctions relevant to this double division of logic: (1) the theolo-
gians were not primarily interested in logic; (2) some of them
elaborated logical doctrines of special importance for theology; an
example is the doctrine of analogy of Thomas Aquinas.
Thus in the Middle Ages we find essentially only one logic.
Exceptions only occur where epistemological or ontological problems
exert an influence, as in the determination of the notion of logic
itself, and in the assigning of denotations. Everywhere else we find a
unified logic, developing organically. The very multiplicity of medie-
val views about extra-logical matters supports the thesis that
formal logic is independent of any special philosophical position on
the part of individual logicians.
E. METHOD
Our insufficient knowledge of the period makes it impossible to
write a history of the evolution of its logic. A historical presentation
would be possible for a few problems only, and even for those only
for isolated spaces of time. The justification of this chapter in a
work on the history of logical problems lies in the fact that, while
un-historical in itself, it does to some extent exhibit one stage in
the general development of logic.
Two questions are raised by the choice of problems for discussion.
The present state of research makes it likely that we are not ac-
quainted with them all. In order not to miss at least the essentials,
we have made great use of the Logica Magna of Paul of Venice, which
expressly refers to all contemporary discussions and may rank as a
veritable Summa of 14th century logic. Paul's range of problems
has been enlarged by some further questions from other authors.
The second difficulty is posed by those logical problems which
overlap epistemology and methodology. Aristotle and the thinkers
of the Megarian-Stoic school envisage them in a fairly simple way,
but scholastic conceptions and solutions are much more complicated.
In order not to overstep our limits too far, these matters will be
touched on only very superficially.
151
SCHOLASTIC LOGIC
F. CHARACTERISTICS
A survey of the logical problems dealt with by the Scholastics
clearly shows that they fall into two classes : on the one hand there
are the ancient ones, Aristotelian or Megarian-Stoic, concerning
e.g. categorical and modal syllogistic, hypothetical syllogisms (i.e.
Stoic arguments) etc. The rest, on the other hand, are either quite
new, or else presented in so new a guise as no longer to remind one
of the Greeks. Conspicuous in this class are the doctrines of 'proper-
ties of terms', of supposition, copulation, appellation and amplia-
tion, then too the doctrine of consequences which while dependent
on Aristotle's Topics and the Stoics, generalizes the older teaching
and puts it in a new perspective. The same must be said about the
insolubles (§ 35) which treat of the Liar and such-like but by new
methods and in a much more general way.
Generally speaking, whatever the Scholastics discuss, even the
problems of antiquity, is approached from a new direction and by
new means. This is more and more the case as the Middle Ages
progress. There is firstly the metalogical method of treatment.
Metalogical items are indeed to be found in Aristotle (14.85 fif.),
but in Scholasticism, at least in the later period, there is nothing
but metalogic, i.e. formulae are not exhibited but described, so
that in many works, e.g. in the De purilaie artis logicae of Burleigh
not a single variable of the object language is to be found. Even
purely Aristotelian matters such as the categorical syllogism are
dealt with from the new points of view, semiotic and other. In early
Scholasticism a double line of development is detectable, problems
inherited from antiquity being treated in the spirit of the ancient
logicians, as in the commentaries of Albert the Great, and the new
doctrine being developed in the very same work. Later the latter
becomes more and more prominent, so that, as has been said, even
genuinely Aristotelian problems are presented metalogically, in
terms of the doctrine of supposition etc.
In addition, scholastic logic, even by the end of the 13th century,
is very rich, very formalistic and exact in its statement. Some
treatises undoubtedly rank higher than the Organon and perhaps
than the Megarian-Stoic fragments too. The title of Burleigh's
work - 'De purilaie arlis logicae' - suits the content, for here is a
genuinely pure formal logic.
152
I. SEMIOTIG FOUNDATIONS
§26. SUBJECT-MATTER OF LOGIC
To be able to understand what the Scholastics thought logic was
about, one must be acquainted with the elements of their semiotic.
Hence we give first two texts from Peter of Spain followed by one
from Ockham, about sounds and terms.
A. BASIC NOTIONS OF SEMIOTICS
26.01 A sound is whatever is properly perceived by hearing;
for though a man or a bell may be heard, this is only by means
of sound. Of sounds, one is voice, another not voice. Sound-
voice is the same as voice; so voice is sound produced from
the mouth of an animal, formed by the natural organs. . . .
Of voices, one is literate, another not literate. Literate voice
is that which can be written, e.g. 'man'; not literate is that
which cannot be written. Of literate voices one is significant,
another not significant. Significant voice is that which
represents something to the hearing, e.g. 'man' or the groans
of the sick which signify pain. Not significant voice is that
which represents nothing to the hearing, e.g. 'bu', 'ba'. Of
significant voices one signifies naturally, another convention-
ally. Conventionally significant voice is that which represents
something at the will of one who originates it, e.g. 'man'.
Naturally significant voice is that which represents the same
thing to all, e.g. the groans of the sick, the bark of dogs. Of
conventionally significant voices one is simple or not complex,
e.g. a noun or a verb, another composite or complex, e.g. a
speech (oratio). . . .
And it should be known that logicians (dialedicus) posit
only two parts of speech, viz. noun and verb , calling the others
'syncategoremata'.
26.02 Of things which are said, some are said with com-
plexity, e.g. 'a man runs', 'white man'. Others without
complexity, e.g. 'man' by itself, a term that is not com-
plex. ... A term, as here understood, is a voice signifying a
universal or particular, e.g. 'man' or 'Socrates'.
These texts contain doctrine generally accepted in Scholasticism.
Another, no less widely recognized, is excellently summarized by
Ockham, who uses the expression 'terminus conceptus' ('conceived
term') instead of the usual 'terminus mentalis' ('thought term').
153
SCHOLASTIC LOGIC
26.03 It is to be known that according to Boethius . . .
speech is threefold, viz. written, spoken and conceived, this
last having being only in the intellect, so (too) the term is
threefold, viz. written, spoken and conceived. A written term
is part of a proposition written down on some body which
is seen or can be seen by a bodily eye. A spoken term is part
of a proposition spoken by the mouth and apt to be heard
with a corporeal ear. A conceived term is an intention or
affection of the soul, naturally signifying something or con-
signifying, apt to be part of a proposition in thought. . . .
Those are the most important presuppositions for what follows.
B. LOGIC AS A THEORY OF SECOND INTENTIONS
Many early Scholastics give explicit definitions of logic. Disregard-
ing these, we shall proceed to descriptions of the subject-matter of
logic, of which we know two kinds. According to the first it consists
in so-called second intentions. Three series of texts will illustrate the
matter, taken from Thomas Aquinas (13th century), Ockham and
Albert of Saxiony (early and late 14th century respectively).
26.04 Being is two-fold, being in thought (ens rationis) and
being in nature. Being in thought is properly said of those
intentions which reason produces (adinvenit) in things it
considers, e.g. the intention of genus, species and the like,
which are not found among natural objects, but are consequent
on reason's consideration. This kind, viz. being in thought,
is the proper subject-matter of logic.
26.05 The relation which is denoted (importatur) by this
name 'the same' is merely a being in thought, if what is the
same without qualification is meant: for such a relation can
only consist in an ordering by the reason of something to
itself, according to some two considerations of it.
26.06 Because relation has the weakest being of all the
categories, some have thought that it belongs to second
intentions (intelledibus). For the first things understood are
the things outside the soul, to which the intellect is primarily
directed, to understand them. But those intentions (inten-
tiones) which are consequent on the manner of understanding
are said to be secondarily understood. ... So according to
this thesis (positio) it would follow that relation is not among
the things outside the soul but merely in the intellect, like the
154
SEMIOTICS
intention of genus and species and second (i.e. universal)
substances.
Thus according to Thomas the subject-matter of logic is such
'secondarily understood things' or 'second intentions', belonging
to the domain of being in thought, and so lecta. Not all lecla, however,
but a special kind, such as those corresponding to the meaning of
logical constants. It is to be stressed that according to Thomas,
as for the Stoics, the subject-matter of logic is nothing psychical,
but something objective, which yet exists only in the soul.
The nature of second intentions war much debated among
Scholastics, and we know of many different opinions. Ockham says:
26.07 It should first be known that that is called an 'inten-
tion of the soul' which is something in the soul apt to signify
something else. . . . But what is it in the soul which is such
a sign? It must be said that on that point (articulum) there
are various opinions. Some say that it is only something
fashioned by the soul. Others that it is a quality subjectively
existing in the soul, distinct from the act of understanding.
Others say that it is the act of understanding. . . . These
opinions will be examined later. For the present it is enough
to say that an intention is something in the soul which is a
sign naturally signifying something for which it can stand
(supponere) or which can be part of a mental proposition.
Such a sign is twofold. One which is a sign of something
which is not such a sign, . . . and that is called a 'first intention'
such as is that intention of the soul which is predicable of
all men, and similarly the intention predicable of all white-
nesses, and blacknesses, and so on. . . . But a second intention
is that which is a sign of such first intentions, such as are the
intentions 'genus', 'species' and such-like. For as one intention
common to all men is predicated of all men when one says:
'this man is a man', 'that man is a man', and so on of each
one, similarly one intention common to those intentions
which signify and stand for things is predicated of them
when one says: . . . 'stone is a species', 'animal is a species,
'colour is a species' etc.
The same doctrine is further developed by Albert of Saxony:
26.08 'Term of first intention' is the name given to that
mental term which is significative of things not from the
point of view of their being signs. Thus this mental term
'man', or this mental term 'being', or this mental term 'qua-
155
SCHOLASTIC LOGIC
lity', or this mental term 'voice'. Hence this mental term
'man' signifies Socrates or Plato, and not insofar as Socrates
or Plato are signs for other things. . . . But a mental term
which is naturally significative of things insofar as they are
signs is called a 'term of second intention', and if they ceased
to be signs it would not signify them. Of this kind are the
mental terms 'genus', 'species', 'noun', 'verb', 'case of a noun'
etc.
In the last two texts the conception is other than that of Thomas.
Second intentions are there conceived in a purely semantic way;
they are signs of signs, and for Albert signs of signs as such.
Whether Ockham and Albert thought of logic as in any sense a
science of second intentions remains open to question. One might
perhaps give expression to both their views by saying that logic
is a science constructed throughout in a meta-language,* remarking
at the same time that the Scholastics included under 'signs' mental
as well as exterior (written or spoken) signs.
However, one common feature underlies all these fundamental
differences; logic is sharply distinguished from ontology in the
whole scholastic tradition. This is so for Thomas, since its object
is not real things, but second intentions; and for his successors, since
it is expressed not in an object- but in a meta-language.
It should also be noted that in fact the entire practice of medieval
logic corresponds to the Thomist conception of the object of logic,
even though this conception was not the only one. For scholastic
logic essentially consists of two parts: the doctrine of the properties
of terms, and the doctrine of consequences. The properties of terms
are evidently second intentions in the Thomist sense; and one must
think oi consequences in the same light, since the logical relationships
they exhibit (e.g. between antecedent and consequent) are not real
things.
C. FORMAL LOGIC AS A THEORY OF
SYNCATEGOREMATIC EXPRESSIONS
There is a difficulty in adopting the view that we have hypotheti-
cally ascribed to Ockham about the subject matter of logic, in that
it does not achieve a definition of logic as a distinct science, since
every science can be formulated in a meta-language. But we find,
though not explicitly, logic limited to concern with logical form,
which leads to an exact definition of formal logic when this form
is equated with the syncategoremata. Scholastic practice is wholly in
accord with this definition in its cultivation of the corresponding
* I am particularly obliged to Prof. E. Moody for valuable assistance with
these questions.
156
SEMIOTICS
theory of logical form. Three texts about Byncategoremata follow,
one from William of Shyreswood (13th century), one from Ockham
(beginning of 14th) and one from Buridan* (end of 14th).
26.09 To understand propositions one must know their
parts. Their parts are twofold, primary and secondary.
Primary parts are substantival names and verbs; these are
necessary for an understanding of propositions. Secondary
parts are adjectival names, adverbs, conjunctions and prepo-
sitions; these are not essential to the constitution of proposi-
tions.
Some secondary parts are determinations of primary ones
with reference to (ratione) their things (i.e. to which they
refer), and such are not syncategoremata ; e.g. when I say
'white man' 'white' signifies that one of its things, a man, is
white. Others are determinations of primary parts insofar as
these are subjects or predicates; e.g. when I say 'every man
runs', the 'every', which is a universal sign, does not mean
that one of its things, namely a man, is universal, but that
'man' is a universal subject. Such are called 'syncategoremata'
and will be treated (here), as offering considerable difficulties
in discourse.
Ockham affords a development of the same thought:
26.10 Categorematic terms have a definite and certain
signification, e.g. this name 'man' signifies all men, and this
name 'animal' all animals, and this name 'whiteness' all
whitenesses. But syncategorematic terms, such as are 'all',
'no', 'some', 'whole', 'besides', 'only', 'insofar as' and such-
like, do not have a definite and certain signification, nor do
they signify anything distinct from what is signified by the
categoremata. Rather, just as in arabic numeration a zero
(cifra) by itself signifies nothing, but attached to another
figure makes that signify, so a syncategorema properly speak-
ing signifies nothing, but when attached to something else
makes that signify something or stand for some one or more
things in a determinate way, or exercises some other function
about a categorema. Hence this syncategorema 'all' has no
definite significate, but when attached to 'man' makes it
* It is taken from the Consequentiae which is ascribed to Buridan in the early
printed editions, though a letter from Fr. Ph. Boehner informs us that no MS of
this work has yet been found.
157
SCHOLASTIC LOGIC
stand or suppose for all men . . . , and attached to 'stone'
makes it stand for all stones, and attached to 'whiteness'
makes it stand for all whitenesses. And the same is to be held
proportionately for the others, as for that syncategorema
'all', though distinct functions are exercised by distinct
syncategoremata, as will later be shown for some of them.
Evidently, the syncategoremata are our logical constants. That
they determine logical form is expressly and consciously propound-
ed by Buridan (whose text was later adopted almost word for
word by Albert of Saxony: 26.11).
26.12 When form and matter are here spoken of, by the
matter of a proposition or consequence is understood merely
the categorematic terms, i.e. the subject and predicate, to
the exclusion of the syncategorematic * ones attached to
them, by which they are restricted, negated, or divided and
given (trahuntur) a determinate kind of supposition. All else,
we say, belongs to the form. Hence we say that the copula,
both of the categorical and of the hypothetical proposition
belongs to the form of the proposition, as also negations,
signs, the number both of propositions and terms, as well
as the mutual ordering of all the aforesaid, and the inter-
connections of relative terms and the ways of signifying
(modos significandi) which relate to the quantity of the
proposition, such as discreteness**, universality etc. . . .
E.g. . . Since modals have subordinate copulas and so differ
from assertoric propositions, these differ in form ; and by reason
of the negations and signs (signa) affirmatives are of another
form than negatives, and universals than particulars; and by
reason of the universality and discreteness*** of their terms
singular propositions are of another form than indefinites ; by
reason of the number of terms the following propositions are
of different forms: 'man is man' and 'man is ass', as are the
following consequences or hypothetical propositions: 'every
man runs, therefore some man runs' and 'every man runs,
therefore some ass walks about'. Similarly by reason of the
order the following are of different forms: 'every man is
animal', 'animal is every man', and likewise the following
* Reading syncategoremaiicis for categoricis.
* * Reading discretio for descriptio.
* * * See last note.
158
CONTENT
consequences: 'every B is A, therefore some B is A1 and
'every B is A, therefore some A is B' etc. Similarly by rea-
son of the relationship and connection . . . 'the man runs, the
man does not run' is of another form than this: 'the man runs
and the same does not run': since its form makes the second
impossible, but it is not so with the first.
It is easy to establish that scholastic logic has for its object
precisely form so conceived. The doctrine of the properties of terms
treats of supposition, appellation, ampliation and such-like rela-
tionships, all of which are determined in the proposition by syn-
categorematic terms; while the second part of scholastic logic,
comprising the doctrine of the syllogism, consequences etc., treats
of formal consequence, which holds in virtue of the form as de-
scribed.
Expressed in modern terms, the difference between the two
conceptions of logic that have been exemplified, is that the first is
semantic, the second syntactical: for the first uses the idea of
reference, the second determines logical form in a purely structural
way. According to the second the logical constants are the subject-
matter of logic, while on the view of Thomas this object is their
sense. On either view Scholasticism achieved a very clear idea of
logical form and so of logic itself.
D. CONTENT OF THE WORKS
Two kinds of logical works can be distinguished in Scholasticism,
commentaries on Aristotle and independent treatises or manuals.
To begin with, the composition of works even of the second kind
is strongly influenced by the Aristotelian range of problems, at least
in the sense that newer problems are incorporated into the frame-
work of the Organon. It is only gradually that the ever growing
importance of the new problems finds expression in the very con-
struction of the works. We shall show this in some examples collected
for the most part by Ph. Boehner (26.13).
Albert the Great has no independent arrangement; his logic
consists of commentaries on the writings of Aristotle and Boethius.
The chief logical work of Peter of Spain falls into two parts; the
first is markedly Aristotelian and contains the following treatises:
On Propositions (= Hermeneia),
On the Predicables (= Porphyry),
On the Categories (= Categories),
On Syllogisms (= Prior Analytics),
On Loci (= Topics),
Suppositions,
On Fallacies (= Sophistic Refutations).
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SCHOLASTIC LOGIC
In the second part is to be found nothing but the new problematic,
for it is divided into treatises on
Relatives,
Ampliations,
Appellations,
Restrictions,
Distributions.
Two points are notable: that propositions are discussed at the start
(and not in the third place as in Porphyry and the Categories), and
that the doctrine of supposition is inserted before the treatise on
fallacies. That shows how the new problematic began to influence
the older one.
Ockham's Summa is divided in another way:
I. Terms:
1. In general.
2. Predicables.
3. Categories.
4. Supposition.
II. Propositions:
1. Categorical and modal propositions.
2. Conversion.
3. Hypothetical propositions.
III. Arguments:
1. Syllogisms:
a) assertoric,
b) modal,
c) mixed (from the first two kinds),
d) 'exponibilia',
e) hypothetical.
2. Demonstration (in the sense of the Posterior Analytics).
3. Further rules:
a) Consequences.
b) Topics.
c) Obligations.
d) Insolubles.
4. Sophistics.
The general framework here is still Aristotelian, more so even than
with Peter, but the new problematic has penetrated into the
subdivisions. An Aristotelian title often conceals strange material,
as when the chapter on the categories deals with typically scholastic
problems about intentions etc.
160
CONTENT
Walter Burleigh's De purilale artis logicae is divided thus:
I. On terms:
1. Supposition.
2. Appellation.
3. Copulation.
II. (Without title):
1. Hypothetical propositions.
2. Conditional syllogisms.
3. Other hypothetical syllogisms.
Even this small sample shows how the scholastic range of problems
is to the fore.
Albert of Saxony divides his logic in this way:
1. Terms (in general).
2. Properties of terms (supposition, ampliation, appellation).
3. Propositions.
4. Consequences:
a) in general.
b) Propositional consequences.
c) Syllogistic consequences.
d) Hypothetical syllogisms.
e) Modal syllogisms.
f) Topics.
5. Sophistics.
6. Antinomies and obligations.
Here the whole of Aristotelian and Stoic formal logic has been built
into the scholastic doctrine of consequences, while this last is
introduced by discussion of another typically scholastic matter,
the properties of terms.
Finally we consider the division of the Logica Magna of Paul
Nicollet of Venice (ob. 1429), which is probably the greatest syste-
matic work on formal logic produced in the Middle Ages. It falls
into two parts, the first designed to treat of terms, the second of
propositions, though in fact the first contains much about propo-
sitions, and the second includes also the doctrine of consequences
and syllogisms.
Part I :
1. Terms.
2. Supposition.
3. Particles that cause difficulty.
4. Exclusive particles.
5. Rules of exclusive propositions.
6. Exceptive particles.
7. Rules of exceptive propositions.
8. Adversative particles.
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SCHOLASTIC LOGIC
9. 'How'.
10. Comparatives.
11. Superlatives.
12. Objections and counter-arguments.
13. Categorematic 'whole' (totus).
14. 'Always' and 'ever'.
15. 'Infinite'.
16. 'Immediate'.
17. 'Begins' and 'ceases'.
18. Exponible propositions.
19. Propositio officiabilis.
20. Composite and divided sense.
21. Knowing and doubting.
22. Necessity and contingence of future events.
Part II :
1. Propositions (in general).
2.-3. Categorical propositions.
4. Quantity of propositions.
5. Logical square.
6. Equivalences.
7. Nature of the proposition in the square.
8. Conversion.
9. Hypothetical propositions.
10. Truth and falsity of propositions.
11. Signification of propositions.
12. Possibility, impossibility.
13. Syllogisms.
14. Obligations.
15. Insolubles.
Here the treatise on consequences has disappeared, having been
incorporated into that on hypothetical propositions.*
§27. SUPPOSITION
We begin our presentation of scholastic logic with the doctrine
of supposition. This is one of the most original creations of Schola-
sticism, unknown to ancient and modern logic, but playing a
* The following figures will give an idea of the scope of this work. The Logica
Magna occupies 199 folios of four columns each containing some 4600 printed
signs, so that the whole work comprises about 3,650,000 signs. This corresponds
to at least 1660 normal octavo pages, four to five volumes. But the Logica Magna
is only one of four works by Paul on formal logic, the others together being even
more voluminous. None of it is merely literary work, but a pure logic, written in
terse and economical language.
162
SUPPOSITION
central role here. Unpublished research of L. Minio Paluello enables
us to trace its origin to the second hall of the 12th century. By the
middle of the 13th all available sources witness to its being every-
where accepted. Later there appear some developments of detail,
but no essentially new fundamental ideas.
We shall first illustrate the notion of supposition in general, then
proceed to the theory of material and simple supposition, and finally
mention other kinds.
A. CONCEPT OF SUPPOSITION
The notion of supposition is already well defined in Shyreswood,
and distinguished by him from similar 'properties of terms' :
27.01 Terms have four properties, which we shall now
distinguish. . . . These properties are signification, supposition,
copulation and appellation. Signification is the presentation
of a form to the reason. Supposition is the ordering of one
concept (intelledus) under another. Copulation is the ordering
of one concept over another. It is to be noted that supposition
and copulation, like many words of this kind, are proffered
(dicuntur) in two senses, according as they are supposed to be
actual or habitual. Their definitions belong to them according
as they are supposed to be actual. But insofar as they are
supposed to be habitual, 'supposition' is the name given
to the signification of something as subsisting; for what
subsists is naturally apt to be ordered under another. And
'copulation' is the name given to the signification of some-
thing as adjacent, for what is adjacent is naturally apt to be
ordered over another. But appellation is the present attribution
of a term, i.e. the property by which what a term signifies can
be predicated of something by means of the verb 'is'.
It follows that signification is present in every part of
speech, supposition only in substantives, pronouns or sub-
stantival particles; for these (alone) signify the thing as
subsistent and of such a kind as to be able to be set in order
under another. Copulation is in all adjectives, participles and
verbs, appellation in all substantives, adjectives and parti-
ciples, but not in pronouns since these signify substance only,
not form. Nor is it in verbs. . . . None of these three, supposi-
tion, copulation and appellation is present in the indeclinable
parts (of speech), since no indeclinable part signifies substance
or anything in substance.
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SCHOLASTIC LOGIC
Thomas Aquinas speaks in similar fashion:
27.02 The proper sense (ratio) of a name is the one which
the name signifies; . . . But that to which the name is attri-
buted if it be taken directly under the thing signified by the
name, as determinate under indeterminate, is said to be
supposed by the name; but if it be not directly taken under
the thing of the name, it is said to be copulated by the name;
as this name 'animal' signifies sensible animate substance,
and 'white' signifies colour disruptive of sight, while 'man'
is taken directly under the sense of 'animal' as determinate
under indeterminate. For man is sensible animate substance
with a particular kind of soul, viz. a rational one. But it is
not directly taken under white, which is extrinsic to its essence.
27.03 The difference between substantives and adjectives
consists in this, that substantives refer to (ferunt) their
suppositum, adjectives do not, but posit in the substance*
that which they signify. Hence the logicians (sophistae) say
that substantives suppose, adjectives do not suppose but
copulate.
The doctrine implicit in these texts was later expressly formulated
by Ockham:
27.04 (Supposition) is a property belonging to terms, but
only as (they occur) in a proposition.
B. MATERIAL AND FORMAL SUPPOSITION
Shyreswood writes:
27.05 Supposition is sometimes material, sometimes formal.
It is called material when an expression (diciio) stands either
for an utterance (vox) by itself, or for the expression which
is composed of an utterance and (its) significance, e.g. if
we were to say: 'homo' consists of two syllables, 'homo' is
a name. It is formal when an expression stands for what it
signifies.
27.06 The first division of supposition is disputed. For it
seems that kinds not of supposition but of signification are
there distinguished. For signification is the presentation of a
form to the reason. So that where there is different presenta-
Reading substantiam for substantivum.
164
SUPPOSITION
tion there is different signification. Now when an expression
supposes materially it presents either itself or its utterance;
but when formally, it presents what it signifies; therefore it
presents something different (in each case); therefore it
signifies something different. But that is not true, since
expressions by themselves always present what they signify,
and if they present their utterance they do not do this of
themselves but through being combined with a predicate. For
some predicates naturally refer to the mere utterance or to
the expression, while others refer to what is signified. But this
effects no difference in the signification. For the expression as
such, before ever being incorporated in a sentence, already
has a significance which does not arise from its being co-
ordinated with another.
On this question Thomas Aquinas remarks:
27.07 One could object to this (teaching of ours) also,
that verbs in other moods (than the infinitive) seem to be put
as subjects, e.g. if one says: 'I run is a verb'. But it must be
said that the verb 'I run' is not taken formally in this state-
ment (locutio). (i.e.) with its signification referred to a thing,
but as materially signifying the word itself which is taken as a
thing.
The expressions 'suppositio materialis' and 'suppositio formalis'
have also another meaning for Thomas. He sometimes uses the
first for suppositio personalis (cf. 27.23 ff.) and the second for
suppositio simplex (cf. 27.17f.) :
27.08 A term put as subject holds (tenetur) materially,
i.e. (stands) for the suppositum; but put as predicate it
holds formally, i.e. (stands) for the nature signified.
Perhaps this ambiguity accounts for the expression 'formal
supposition', that we have found in Shyreswood and Thomas, later
disappearing, so far as we know, outside the Thomist school.*
Even by Ockham's time supposition is divided immediately into
three kinds:
27.09 Supposition is first divided into personal, simple
and material.
*Fr. Ph. Boehner is to be thanked for the information that this expression
occurs in Chr. Javellus (ob. 1538).
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SCHOLASTIC LOGIC
The two first of those are sub-species of the formal supposition of
Shyreswood and Thomas, which Ockham no longer refers to. His
division is subsequently the usual one, except among the Thomists.
In 27.05 we read of an 'utterance by itself and an 'expression
which is composed of an utterance and (its) significance'. This
distinction is developed at the end of the 15th century by Peter
Tarteret:
27.10 Material supposition is the acceptance of a term for
its non-ultimate significate, or its non- ultimate significates
In which it is to be noticed that significates are two-fold,
ultimate and non-ultimate. The ultimate significate is that
which is ultimately signified by a term signifying conven-
tionally, and ultimately or naturally and properly. But the
non-ultimate significate is the term itself, or one vocally or
graphically similar, or one mentally equivalent. From which
it follows that a vocal or written term is said to signify
conventionally in two ways, either ultimately or non-ulti-
mately. Ultimately it signifies what it is set to signify; but
a vocal term is said to signify conventionally and non-ulti-
mately a synonymous written term; and a written term
is said to signify non-ultimately an utterance synonymous
with it. . . .
From the modern point of view this doctrine reflects our distinc-
tion of language and meta-language, except that in place of two
languages, symbols of one language exercise a two-fold supposition.
Furthermore, the two last-cited texts exhibit the important distinc-
tion between the name of an individual symbol and the name of a
class of equiform symbols. We do not find this in the logistic period
till after 1940.
This distinction first occurs, so far as we know, in St. Vincent
Ferrer* (14th century), as a division of material supposition:
27.11 Material supposition is divided as is formal. One
(kind of material supposition) is common, the other discrete.
* Vincent Ferrer was the greatest preacher of his time. We would add that
Savonarola was also an important logician. A similar link between deep religiDUS
life and a talented interest in formal logic is also to be observed in Indian culture
especially among the Buddhists. This would seem to be a little known and as yet
unexplained phenomenon. The authenticity of Vincent Ferrer's philosophical
opuscules De Suppositionibus and De imitate universalis has been challenged so
far as we know only by S. Brettle (vid. Additions to Bibliography 3.98). To his
p. 105 note 3 should be added a, here relevant, reference to p. 33 note 10. M. G.
Miralles (vid. Additions) summarizes the arguments for and against, and
concludes with M. Gorce (vid. Additions): 'L'authenticite des deux 6crits n'a ete
jamais mise en doute. Le temoignage du contemporain Ranzzano suffit a la
prouver.'
166
SUPPOSITION
It is discrete if the term or utterance stands determinately
for a suppositum of its material significate. And thus discrete
material* suppositum occurs in three ways. In one way
through the utterance or term itself, as when one asks:
'What is it you want to say?' and the other answers: 'I say
"buf" and '"baf" is said by me', (then) the subject of this
proposition supposes materially and discretely since it stands
for the very utterance numerically identical (with it) (cf.
11.11). This becomes more evident if names are assigned to
the individual terms in such a way that as this name 'man' *
signifies this individual man so this name lA' signifies that
individual word 'buf' and '£' the other ('baf'). And then if
it is said: lA is an utterance' or 'A is said by me', the
subject supposes materially and discretely, as in the proposi-
tion 'Socrates runs' the subject supposes formally and dis-
cretely.
27.12 It occurs secondly through a demonstrative name
(nomen) demonstrating an utterance or singular term, as
when the utterance of the term 'man' is written somewhere
and one says, with reference to this utterance: 'That is a
name'. Then the subject of the proposition supposes materially
for that which it demonstrates.
It occurs in a third way through a term. . . ., which is
determined by a demonstrative pronoun, as when it is said
of the written utterance 'man': 'this "man" is a name' or
'this utterance is a name'.
And each of these ways . . . can be varied by natural,
personal or simple supposition, as was said about singular
formal supposition.
Common (communis) material supposition is when the
utterance or term stands indeterminately for its material
signification, as when it is said: "people" is written' the sub-
ject of this proposition stands indeterminately for this term
'people', or (in another example) for some other (term). I
do not say that in the proposition '"people" is written' or in
some other such that the supposition is indeterminate, but
that the subject is indeterminate and is taken indeterminately
Material common supposition is divided into natural,
personal and simple supposition, like common formal supposi-
* Reading materialis for formalis.
** Reading homo for primo.
167
SCHOLASTIC LOGIC
tion. . . . An example of personal : ' "man" is heard', ' "man" is
written ',' "man" is answered'. An example of simple
(supposition): '"man" is a species of utterance', '"man" is
conceived', '"man" is said by this man'; and so on in many
other cases as everyone can see for himself.
So material supposition is divided just as is formal. These texts
exhibit scholastic semantics at its best. This accuracy of analysis
is the more astonishing when one remembers that the distinction
mentioned in the introduction to 27.11 remained unknown, not
only to the decadent 'classical' logicians, but also to mathematical
ones for nearly a century.
It should also be noticed that in the text of Tarteret just cited, a
distinction occurs which cannot be expressed in contemporary
terms. The Scholastics distinguished, as has already be said above
(26.03) three inter-related kinds of sign: graphical, vocal and
psychic, and a materially supposing graphical sign can stand either
for itself (or its equiforms) or for the corresponding vocal or psychic
sign.
Burleigh has another division, parallel to that between material
and formal supposition:
27.13 The tenth rule is: that on every act that is accom-
plished there follows the act that is signified, and conversely.
For it follows: 'man is an animal, therefore "animal" is
predicated of "man"', for the verb 'is' accomplishes predica-
tion, and this verb 'is predicated' signifies predication, and
syncategorematic particles accomplish acts, and adjectival
verbs signify such acts. E.g. the sign 'all' accomplishes
distribution, and the verb 'to distribute' signifies distributions;
the particle 'if exercises consequence, and this verb 'it follows'
signifies consequence.
It was said above that this distinction runs parallel with that
between formal and material supposition, for it could easily be
translated into it. But Burleigh would not seem to be thinking of
these suppositions here; by 'the act signified' he means not words,
but their significates. For in his example, the word 'animal' is not
predicated of the word 'man', but what the first signifies is predi-
cated of that for which the second supposes.
C. SIMPLE SUPPOSITION
Along with the idea of material supposition, that of simple
(simplex) supposition is an interesting scholastic novelty. On this
subject we can limit ourselves to the 13th century, and mainly to
168
SUPPOSITION
Peter of Spain. First we shall give some of his general divisions of
formal supposition :
27.14 One kind of supposition is common, another discrete.
Common supposition is effected by a common term such as
'man'. Again of common suppositions one kind is natural,
another accidental. Natural supposition is the taking of a
common term for everything of which it is naturally apt to be
predicated, as 'man' taken by itself naturally possesses
supposition for all men who are and who have been and who
will be. Accidental supposition is the taking of a common
term for everything for which its adjunct requires (it to be
taken). E.g. 'A man exists'; the term 'man' here supposes for
present men. But when it is said: 'a man was', it supposes for
past men. And when it is said : 'a man will be', it supposes for
future ones, and so has different suppositions according to the
diversity of its adjuncts.
Later on we also meet with an 'improper' (27.15) and a 'mixed'
(27.16) supposition. The first simply consists in the metaphorical
use of a term. The second was introduced to elucidate the function
of terms of which one part supposed in one way, another in another.
From the logical point of view these are not very important ideas.
Of greater importance is Peter's continuation :
27.17 Of accidental suppositions one is simple, another
personal. Simple supposition is the taking of a common term
for the universal thing symbolized (figurata) by it, as when it
is said : 'man is a species' or 'animal is a genus', the term 'man'
supposes for man in general and not for any of its inferiors,
and similarly in the case of any common term, as 'risible is a
proprium', 'rational is a difference'.
27.18 Of simple suppositions one belongs to a common
term set as subject, as 'man is a species'; another belongs to a
common term set as an affirmative predicate, as 'every man
is an animal'; the term 'animal' set as a predicate has simple
supposition because it only supposes for the generic nature;
yet another belongs to a common term put after an exceptive
form of speech, as 'every animal apart from man is irrational'.
The term 'man' has simple supposition. Hence it does not
follow: 'every animal apart from man is irrational, therefore
every animal apart from this man (is irrational)', for there
is there the fallacy of the form of speech (cf. 11.19), when
passage is made from simple to personal supposition. Similarly
169
SCHOLASTIC LOGIC
here: 'man is a species, therefore some man (is a species)'. In
all such cases passage is made from simple to personal sup-
position.
27.19 But that a common term put as predicate is to be
taken with simple supposition is clear when it is said : 'of all
contraries there is one and the same science', for unless the
term 'science' had simple supposition there would be a fallacy.
For no particular science is concerned with all contraries;
medicine is not concerned with all contraries but only with
what is healthy and what is sick, and grammar with what is
congruous and incongruous, and so on.
This is to be compared with the text of Thomas cited above
(27.08). The following text from him from a theological context
expresses the matter clearly:
27.20 The proposition homo fadus est Deus . . . can be
understood as though fadus determines the composition, so
that the sense would be: 'a man is in fact God', i.e. it is a fact
that a man is God. And in this sense both are true, homo fadus
est Deus and Deus fadus est homo. But this is not the proper
sense of these propositions (locutionum), unless they were to be
so understood that 'man' would have not personal but simple
supposition. For although this (concrete) man did not become
God, since the suppositum of this, the person of the Son of
God, was God from eternity, yet man, speaking universally,
was not always God.
This text has the further importance that it may suggest the rea-
son why the Scholastics spoke of 'personal' supposition, this being
the function exercised by a term in standing for individuals or an
individual (suppositum). For this recalls to the mind of a Scholastic
the famous theological problem of the person of Christ, as in 27.20.
The essentials of the scholastic doctrine of simple supposition
may be summed up thus: in the proposition lA is B', the subject 'A'
has of itself personal supposition, i.e. it stands for the individuals,
but the predicate lB' has simple supposition, i.e. it stands either for
a property or a class. But one can also frame propositions in which
something is predicated of such a property or class, and then the
subject must have simple supposition. It can be seen that this
doctrine deals with no less a subject than the distinction between
two logical types, the first and second (cf. 48.21).
These simple but historically important facts are complicated by
the scholastic development of two other problems along with this
doctrine. They are (1) the problem of analysing propositions,
170
SUPPOSITION
whether they should be understood in a purely extensions! fashion,
or with extensional subject and intensional predicate. Thomas and
Peter, in the texts cited, adopt the second position. We shall treat
this problem a little more explicitly in a chapter on the analysis
of propositions (29.02-04). Then (2) there is the problem of the
semantic correlate of a term having simple supposition. This is a
very difficult philosophical problem, and the Scholastics were of
varying opinions about its solution. In 27.18 Peter seems to think
that a term with simple supposition stands for the essence (nature)
of the object. On the other hand Ockham and his school hold that
the semantic correlate of such a term is simple, 'the intention of the
soul':
27.21 A term cannot have simple or material supposition
in every proposition but only when ... it is linked with another
extreme which concerns an intention of the soul or an utterance
or something written. E.g. in the proposition 'a man is running'
the 'man' cannot have simple or material supposition, since
'running' does not concern either an intention of the soul, nor
an utterance nor something written. But in the proposition
'man is a species' it can have simple supposition because
'species' signifies an intention of the soul.
In this and similar texts (27.22) it is of logical interest that
Ockham and his followers were apparently trying to give an exten-
sional interpretation even to terms having simple supposition;
their correlates would be (concrete) intentions.
After Buridan there were in the Middle Ages, as at the beginning
of the 20th century, some logicians who equated simple and material
supposition. Paul of Venice gives that information :
27.23 Simple supposition is distinct from material and
personal ; some say otherwise, and make no distinction between
simple and material supposition. But (wide) it is evident that
the subject does not suppose materially when it is said: 'the
divine essence is inwardly communicable'.
D. PERSONAL SUPPOSITION
The most usual supposition of a term is personal. As Ockham
says:
27.24 It is also to be noticed that in whatever proposition
it be put, a term can always have personal supposition, unless
it be restricted to some other by the will of those who use it.
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SCHOLASTIC LOGIC
We give the definition and divisions of this kind of supposition
according to Peter of Spain, whose text contains the essentials of
the doctrine that remained standard till the end of the scholastic
period.
27.25 Personal supposition is the taking of a common term
for its inferiors, as when it is said 'a man runs', the term 'man'
supposes for its inferiors, viz. for Socrates and for Plato and
so on.
27.26 Of personal suppositions one kind is determinate,
another confused. Determinate supposition is the taking of a
common term put indefinitely or with the sign of particularity,
as 'a man runs' or 'some man runs', and both are called
'determinate' because although in both the term 'man'
supposes for every man, whether running or not, yet they are
true only for one man running. For it is one thing to suppose
(for things), and another to render the proposition true for
one of them*. But as has been said, the term 'man' supposes
for all whether running or not, yet renders the propositions
true only for one who is running. But it is clear that the
supposition is determinate in both (propositions), because
when it is said: 'An animal is Socrates, an animal is Plato,
and so on, therefore every animal is every man', this is the
fallacy of the form of speech (proceeding) from a number of
determinates to one (cf. 11.19 and 27.18). And so a common
term put indefinitely has determinate supposition, and
similarly if it has the sign of particularity.
27.27 But confused supposition is the taking of a common
term for a number of things by means of the sign of univer-
sality, as when it is said: 'every man is an animal', the term
'man' is taken for a number by means of the sign of uni-
versality, being taken for each of its individuals.
Subsequent division of confused supposition into that which is
confused by the requirements of the sign (necessitate signi) and that
which is confused by the requirements of the thing (rei) (27.28) is
shortly after rejected by Peter. He gives a further division of perso-
nal supposition:
27.29 Of personal supposition one kind is restricted,
another extended (ampliata).
* Reading praedictis for praedicatis.
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AMPLIATION
E. INTERPRETATION IN MODERN TERMS
If we ask how the expression 'supposition' is to be rendered in
modern terms, we have to admit that it cannot he. 'Supposition'
covers numerous semiotic functions for which we now have no
common name. Some kinds of supposition quite clearly belong to
semantics, as in the case of both material suppositions, and personal ;
others again, such as simple supposition and those into which
personal supposition is subdivided, are as Moody has acutely remark-
ed (27.30), not semantical but purely syntactical functions.
The most notable difference between the doctrine of supposition
and the corresponding modern theories lies in the fact that while
contemporary logic as far as possible has one sign for one function,
e.g. a sign for a word, another for the word's name, one for the word
in personal, another for it in simple supposition, the Scholastics
took equiform signs and determine their functions by establishing
their supposition. And this brings us back to the fundamental
difference already remarked on between the two forms of formal
logic; scholastic logic dealt with ordinary language, contemporary
logic develops an artificial one.
§28. AMPLIATION, APPELLATION, ANALOGY
Among the other properties of terms three that seem to be of parti-
cular interest for formal logic will be illustrated with some texts,
viz. ampliation, appellation and analogy.
A. AMPLIATION
Peter of Spain writes:
28.01 Restriction is the narrowing of a common term from
a wider (maiore) supposition to a narrower, as when it is said
'a white man runs' the adjective 'white' restricts 'man' to
supposing for white ones. Ampliation is the extension (exiensio)
of a common term from a narrower supposition to a wider, as
when it is said 'a man can be Antichrist' the term 'man'
supposes not only for those who are now, but also for those who
will be. Hence it is extended to future ones. I say 'of a common
term' because a discrete term is neither restricted nor extended.
One kind of ampliation is effected by a verb, as by the
verb 'can', e.g. 'a man can be Antichrist'; another is effected
by a name, e.g. 'it is possible that a man can be Antichrist' ;
another by a participle, e.g. 'a man is able (potens) to be
Antichrist' ; another by an adverb, e.g. 'a man is necessarily an
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SCHOLASTIC LOGIC
animal'. For (in the last) 'man' is extended not only for the
present but also for the future. And so there follows another
division of ampliation : one kind of ampliation being in respect
of supposita, e.g. 'a man can be Antichrist', another with
respect to time, e.g. 'a man is necessarily an animal', as has
been said.
Essentially the same doctrine but more thoroughly developed is
found at the end of the 14th century in Albert of Saxony:
28.02 Ampliation is the taking of a term for one or more
things beyond what is actually the case: for that or those
things for which the proposition indicates (denotat) that it is
used. Certain rules are established in this respect:
28.03 The first is this: every term having supposition in
respect of a verb in a past tense is extended to stand for what
was, e.g. when it is said: 'the white was black', 'the white' is
taken in this proposition not only for what is white but for
what was white.
28.04 Second rule : a term having supposition in respect of
a verb in a future tense is extended to stand for what is or
will be. . . .
28.05 Third rule : every term having supposition with
respect to the verb 'can' is so extended as to stand for what is
or can be. E.g. 'the white can be black' means that what is
white or can be white, can be black. . . .
28.06 Fourth rule : A term having supposition in respect of
the verb 'is contingent' is extended to stand for what is or
can contingently be (contingit esse). And that is Aristotle's
opinion in the first book of the Prior {Analytics). . . .
28.07 Fifth rule: A term subjected in a proposition in
respect of a past participle, even though the copula of this
proposition is a verb in the present, is extended to stand for
what was. . . . E.g. in the proposition 'a certain man is dead'
the subject stands for what is or has been.
28.08 Sixth rule : In a proposition in which the copula is in
the present, but the predicate in the future, the subject is
extended to stand for what is or will be. E.g. 'a man is one
who will generate'; for this proposition indicates that one
who is or will be a man is one who will generate.
28.09 Seventh rule: If the proposition has a copula in the
present and a predicate that includes the verb 'can', as is the
case with verbal names ending in '-ble' ('-ibile'), then the sub-
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APPELLATION
ject is extended to stand for what is or can be, e.g. when it is
said: 'the man is generable'. For this is equivalent (valet) to:
'the man can be generated' in which 'man' is extended,
according to the third rule, to stand for what is or can be. . . .
28.10 Eighth rule: all verbs which, although not in the
present, have it in their nature to be able to extend to a future,
past or possible thing as to a present one, extend the terms to
every time, present, past and future. Such e.g. are these:
'I understand', 'I know', 'I am aware', 'I mean (significo)' etc.
20.11 Ninth rule: the subject of every proposition de
necessario in the divided sense (cf. § 29, D.) is extended to
stand for what is or can be. E.g. 'every B is necessarily A';
for this is equivalent to (valet dicere) 'Whatever is or can be
B, is necessarily A'. . . .
28.12 Tenth rule: if no ampliating term is present in a
proposition, its subject is not extended but this proposition
indicates that (the subject stands) only for what is.
This text is a fine example of scholastic analysis of language. It
introduces a notable enlargement of the doctrine of supposition,
dividing the objects for which a term may stand into three tem-
poral classes to which is added the class of possible objects. It
can readily be seen that this doctrine makes an essential contri-
bution to the problem of the so-called void class, since the expres-
sion 'void class' receives as many different denotata as there are
kinds of ampliation. This can be compared with the modern me-
thods of treating the problem (cf. § 46, A and B).
Albert's seventh, eighth and ninth rules also contain an analysis
of modal propositions, but this subject will be considered in greater
detail below (§ 33).
B. APPELLATION
Closely connected with ampliation is the so-called appellation,
also relevant to the problem of the void class. The theory of it was
already well developed in the 13th century, was further enlarged in
the 14th when there were various theories different from that of the
13th.* We cite two 13th century texts, one from Peter of Spain
and one from Shyreswood:
28.13 Appellation is the taking of a term for an existent
thing. I say 'for an existent thing' since a term signifying a
non-existent has no appellation, e.g. 'Caesar' or 'Anti-christ'
* For information on these points and much other instruction on the doc-
trines of supposition and appellation I am obliged to Prof. E. Moody.
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SCHOLASTIC LOGIC
etc. Appellation differs from supposition and signification in
that appellation only concerns existents, but supposition and
signification concern both existents and non-existents, e.g.
'Antichrist' signifies Antichrist and supposes for Antichrist
but does not name (appellai) him, whereas 'man' signifies man
and naturally supposes for existent as well as non-existent
men but only names existent ones.
Of appellations one kind belongs to common terms such as
'man', another to singular terms such as 'Socrates'. A singular
term signifies, supposes and names the same thing, because it
signifies an existent, e.g. 'Peter'.
Further, of appellations belonging to common terms one
kind belongs to a common term (standing) for the common
thing itself, as when a term has simple supposition, e.g. when
it is said : 'man is a species' or 'animal is a genus' ; and then the
common term signifies, supposes and names the same thing,
as 'man' signifies man in general and supposes for man in
general and names man in general. Another kind belongs to a
common term (standing) for its inferiors, as when a common
term has personal supposition, e.g. when it is said : 'a man runs'.
Then 'man' does not signify, suppose and name the same
thing; because it signifies man in general and supposes for
particular men and names particular existent men.
28.14 Supposition belongs to (inest) a term in so far as it
is under another. But appellation belongs to a term in so far
as it is predicable of its (subordinate) things by means of the
verb 'is'. . . . Some say therefore that the term put as subject
supposes, and that put as predicate names. ... It should also
be understood that the subject-term names its thing, but not
qua subject. The predicate-term on the other hand names it
qua predicate.
The following from Buridan may serve as an example of 14th
century theories:*
28.15 First it is to be understood that a term which can
naturally suppose for something names all that it signifies
or consignifies unless it be limited to what it stands for. . . .
E.g. 'white' standing for men names whiteness, and 'great'
greatness, and 'father' the past (act of) generation and someone
* These texts were communicated by Prof. E. Moody who also pointed out
their great importance. He is to be thanked also for the main lines of the commen-
tary.
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ANALOGY
else whom the father has generated, and 'the distant' names
that from which it is distant and the space (dimensionem)
between them by which it is made distant. . . .
28.16 A term names what it names as being somehow
determinant (per modum adiacenlis aliquo modo) or not
determinant of that for which it stands or naturally can
stand. . . .
Thirdly it is to be held that according to the different
positive kinds of determination of the things named - the things
for which the term stands - there are different kinds of
predication, such as how, how many, when, where, how one is
related to another, etc. It is from these different kinds of
predication that the different predicaments are taken . . .
(cf. 11.15).
28.17 Appellative terms name differently in respect of an
assertoric verb in the present and in respect of a verb in the
past and in the future, and in respect of the verb 'can' or of
'possible'; since in respect of a verb in the present the appel-
lative term - provided there is not ampliative term - whether
it be put as subject or predicate, names its thing as something
connected with it in the present, for which the term can
naturally stand, and as connected with it in this or that man-
ner, according to which it names.
This is a different doctrine from that of the 13th century, and
seems to be of the highest importance. For according to it a term
does not name what it stands for but something related to it by,
it would seem, any relation. Buridan says this expressly for the term
'distant'. If A is distant from B 'distant' does not name A, but
precisely B. That indicates a clear notion of relation-logic. Where
we should write 'relation', Buridan has adiacentia. Especially im-
portant is 28.16 where Buridan goes so far as to say that absolute
terms are definable by relations, an idea corresponding to the relative
descriptions of 47.20. Some interesting results would follow from the
detailed working out of the basic notions of this text, e.g. a theory
of plural quantification, but we have no knowledge of this being
done in the Middle Ages.
C. ANALOGY
In the present state of research it is unfortunately impossible
to present the scholastic theory of meaning with any hope of
doing justice even only to its essentials. However, we shall treat of
a further important point in this field, the theory of analogy. This
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SCHOLASTIC LOGIC
is of direct relevance to formal logic, and fairly well explored. A
single text from Thomas Aquinas will suffice:
28.18 Nothing can be predicated univocally of God and
creatures; for in all univocal predication the sense (ratio) of
the name is common to both things of which the name is
univocally predicated . . . and yet one cannot say that what is
predicated of God and creatures is predicated purely equivo-
cally. ... So one must say that the name of wisdom is predi-
cated of God's wisdom and ours neither purely univocally
nor purely equivocally, but according to analogy, by which is
just meant: according to a proportion. But conformity
(convenientia) according to a proportion can be twofold, and
so a twofold community of analogy is to be taken account of.
For there is a conformity between the things themselves which
are proportioned to one another in having a determinate
distance of some other relationship (habitudinem) to one ano-
ther, e.g. (the number) 2 to unity, 2 being the double. But we
also sometimes take account of conformity between two things
which are not mutually proportioned, but rather there is a
likeness between two proportions; e.g. 6 is conformed to 4
because as 6 is twice 3 so 4 is twice 2. The first conformity then
is one of proportion, but the second of proportionality. So it is
then that according to the first kind of conformity we find
something predicated analogically of two things of which
one has a relationship to the other, as being is predicated of
substance and of accident owing to the relationship which
substance and accident have (to one another), and health is
predicated of urine and animals, since urine has some relation-
ship * to the health of animals. But sometimes predication is
made according to the second kind of conformity, e.g. the
name of sight is predicated of corporeal sight and of intellect,
because as sight is in the eye, so intellect is in the mind.
This is about the clearest text of the many in which Thomas
Aquinas speaks of analogy (28.19). It has been only too often
misunderstood, but deserves fairly thorough discussion from the
historian of logic because of its historical as well as systematic
significance. We therefore draw attention to the following points:
This text deals explicitly with a question of semantics - Thomas
speaks of names - and it is noteworthy that he himself, like his best
commentator Cajetan (28.20), almost always considers analogy as
* Reading habitudinem for similitudinem.
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ANALOGY
'of names'. Of course he does not mean mere utterances, but mean-
ingful words, in accordance with the scholastic usage illustrated
previously.
Now our text speaks of three classes of names: univocal, equivo-
cal and analogous names. The last are intermediate between the
two first. The class of analogous names falls into two sub-classes:
those analogous according to a proportion, and those according to
proportionality. Both these divisions originate with Aristotle
(10.29 and 10.31), but the hasty indications of the Nichomachean
Ethics are here developed into a systematic logical doctrine.
While the thomist doctrine of the first class of analogous names
is here only of interest as showing an attempt to formalize the rules
of their use, the theory of the second class, i.e. of names analogous
according to proportionality, is nothing less than a first formulation
of the notion of isomorphy (cf. 47.41). That this is so can be seen as
follows :
Let us note first that according to the text an analogous name of
the second kind always refers to a relation or relata defined by one.
Certainly something absolute is also implied by each of the subjects
in such an analogy, but this is precisely different in each, and in
that respect the name is equivocal. The community of reference
consists only in regard to certain relations.
But it is not a matter of just one relation, rather of two similar
ones. This is explicit in the text, only the example (6:3 = 4:2) is
misleading since we have there an identity of two relations. That
Thomas is not thinking of such is shown by the illustrations, first in
the domain of creatures (sight: eye — intellect: mind), then in God
(divine being: God - creaturely being: creature). The ruling idea is
then of a relation of similarity between two relations.
This relation between relations is such as to allow inference from
what we know about one to something about the other, though at
the same time we have the assertion : 'we cannot know what God is'
(28.21). The apparent contradiction disappears when it is realized
that we are dealing with isomorphy. For this does in fact allow one
to transfer something from one relation to another, without afford-
ing any experience of the relata.
The use of a mathematical example is noteworthy, taken more-
over from the only algebraic function then known. This is not only to
be explained by the mathematical origin of the doctrine of analogy
in Aristotle, but also perhaps by a brilliant intuition on the part of
Aquinas who dimly guessed himself to be establishing a thesis about
structure. In any case the text is of the utmost historical importance
as being the first indication of a study of structure, which was to
become a main characteristic of modern science.
179
§ 29. STRUCTURE AND SENSE OF PROPOSITIONS
A. DIVISION OF PROPOSITIONS
We give first a text of Albert of Saxony which summarizes the
commonly received scholastic doctrine of the kinds of atomic
(categorical) propositions :
29.01 Of proposition some are categorical, others hypo-
thetical. But some of the categorical are said to be hypothetical
in signification, such as the exclusive, exceptive and redupli-
cative propositions, and others besides.
Then of the categorical propositions that are not equi-
valent to the hypothetical in signification - such as 'man is an
animal' and such-like - some are said to be assertoric (de
inesse) or of simple inherence; others are said to be modal or
of modified inherence. . . .
Again of categorical propositions of simple inherence some
have ampliative subjects, as 'a man is dead', 'Antichrist will
exist', the others do not have ampliative subjects, as 'man is an
animal', 'stone is a substance' etc.
Again, of categorical propositions of simple inherence
with ampliative subjects, some concern the present, others the
past, others the future. . . .
Again, of categorical propositions about the present some
are de secundo adiacente, others de tertio adiacente. An example
of the first : 'man exists' ; of the second : 'man is an animal'.
Again, of categorical propositions some have a non-com-
pound extreme (term) (de extremo incomplexo), as 'man is an
animal'; others have a compound extreme, as 'man or ass is
man or ass'.
B. ANALYSIS OF PROPOSITIONS
Here we assemble a few aspects of the scholastic analysis of
propositions. To begin with, this text of Thomas Aquinas, followed
by one from Ockham, about the general structure:
29.02 In every true affirmative proposition the subject
and predicate must signify somehow the same thing in reality
but in different senses (diversum secundum rationem). And this
is clear both in propositions with accidental predicate and in
those with substantial. For it is evident that 'man' and
'white' are identical in suppositum and differing in sense, for
the sense of 'man' is other than the sense of 'white'. And
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PROPOSITIONS
likewise when I say: 'man is an animal', for that same thing
which is man is truly an animal. For in one and the same
suppositum there is both the sensible nature, after which it is
called 'animal', and the rational nature, after which it is called
'man'. So that in this case too the predicate and subject are
identical as to suppositum, but differing in sense. But this is
also found in a way in propositions in which something (idem)
is predicated of itself, inasmuch as the intellect treats as
suppositum (trahit ad partem suppositi) what it posits as
subject, but treats as form inhering in the suppositum what it
posits as predicate. Hence the adage, that predicates are
taken formally and subjects materially (cf. 27.08). To the
difference in sense there corresponds the plurality of predi-
cate and subject; but the intellect signifies the real identity by
the composition (of the two).
We have here actually two analyses of propositions. First an
extensional one, which seems to have become classical in later
Scholasticism. It can be reproduced thus: the proposition 'S is P'
is to be equated with the product of the following propositions:
(1) there is at least one x such that both lS' and 'P' stand (suppose)
for x, (2) there is a property / such that 'S' signifies /, (3) there is a
property g such that 'P' signifies g, (4) both / and g belong to x.
In the second analysis the subject is conceived as extensional, the
predicate as intensional. The proposition 'A = A' can be interpreted :
(1) there is an x such that 'A' stands for x, (2) there is a property /
such that 'A' signifies /, (3) / belongs to x. This analysis is applied in
the text to a special kind of proposition, asserting an identity, but
can evidently be applied generally.
Ockham gives another analysis :
29.03 It is to be said that it is not required for the truth of
a singular proposition, which is not equivalent to many
propositions, that subject and predicate should be really
identical, nor that the predicated reality should be in the
subject, nor that it should really inhere (insit) in the subject,
nor that it should be really, extra-mentally, united to the
subject. E.g. it is not required for the truth of this: 'that one
is an angel', that the common term 'angel' should be in reality
the same as what is posited as subject, nor that it should
really inhere in it, nor anything of that kind, - but it is
sufficient and necessary that subject and predicate should
suppose for the same thing. And so in this : 'this is an angel', if
subject and predicate suppose for the same thing, the pro-
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SCHOLASTIC LOGIC
position is true. And so it is not indicated (denotatur) that this
has angelicity, or that angelicity is in it, or anything of this
kind, but it is indicated that this is truly an angel, not that it
is that predicate, but that it is that for which the predicate
supposes.
An important text, but not readily intelligible to a modern reader.
A possible, though not the only possible interpretation is this: it is
sufficient and necessary for the truth of a proposition of this kind
that the extension of subject and predicate should coincide. If that
is so, he means to say that the predicate is not to be taken intension-
ally, but extensionally like the subject, as in Thomas's first ana-
lysis in 29.02. Then Ockham gives a radically extensional inter-
pretation of propositions.
The next text shows that what was said in 29.03 holds for other
kinds of proposition as well:
29.04 For the truth of such (i.e. indefinite or particular
propositions) it suffices that the subject and predicate stand
for the same thing, if the proposition is affirmative.
C. ANALYSIS OF MODAL PROPOSITIONS: DICTUM AND MODUS
In the middle of the 13th century there arose a generally accepted
doctrine about the structure of modal propositions. It is to be found
in Albert the Great (29.05), Shyreswood (29.06), Peter of Spain (29.07),
and in the Summa Toiius Logicae (29.08). On account of its charac-
teristic formalism we quote a youthful opusculum of Thomas
Aquinas:
29.09 Since the modal proposition gets its name from
'modus', to know what a modal proposition is we must know
what a modus is. Now a modus is a determination of something
effected by a nominal adjective determining a substantive,
e.g. 'white man', or by an adverb determining a verb. But it
is to be known that modes are threefold, some determining
the subject of a proposition, as 'a white man runs', some deter-
mining the predicate, as 'Socrates is a white man' or 'Socrates
runs quickly', some determining the composition of the
predicate with the subject, as 'that Socrates is running is
impossible', and it is from this alone that a proposition is said
to be modal. Other propositions, which are not modal, are said
to be assertoric (de inesse).
The modes which determine the composition are six: 'true',
'false', 'necessary', 'possible', 'impossible' and 'contingent'.
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PROPOSITIONS
But 'true' and 'false' add nothing to the signification of asser-
toric propositions; for there is the same significance in
'Socrates runs' and it is true that Socrates runs' (on the one
hand), and in 'Socrates is not running' and 'it is false that
Socrates is running' (on the other). This does not happen with
the other four modes, because there is not the same significance
in 'Socrates runs' and 'that Socrates runs is impossible (or
necessary)'. So we leave 'true' and 'false' out of consideration
and attend to the other four.
Now because the predicate determines the subject and not
conversely, for a proposition to be modal the four modes
aforesaid must be predicated and the verb indicating com-
position must be put as subject. This is done if an infinitive is
taken in place of the indicative verb in the proposition,
and an accusative in place of the nominative. And it (the
accusative and infinitive clause) is called 'dictum', e.g. of the
proposition 'Socrates runs' the dictum is 'that Socrates runs'
(Socratem currere). When then the dictum is posited as subject
and a mode as predicate, the proposition is modal, e.g. 'that
Socrates runs is possible'. But if it be converted it will be
assertoric, e.g. 'the possible is that Socrates runs'.
Of modal propositions one kind concerns the dictum,
another concerns things. A modal (proposition) concerning
the dictum is one in which the whole dictum is subjected and
the mode predicated, e.g. 'that Socrates runs is possible'. A
modal (proposition) concerning things is one in which the
mode interrupts the dictum, e.g. 'for Socrates running is
possible' (Socratem possibile est currere). But it is to be known
that all modals concerning the dictum are singular, the mode
being posited as inherent in this or that as in some singular
thing. But . . . modals concerning things are judged to be
universal or singular or indefinite according to the subject of
the dictum, as is the case with assertoric propositions. So
that 'for all men, running is possible' is universal, and so
with the rest. It should further be known that modal pro-
positions are said to be affirmative or negative according
to the affirmation or negation of the mode, not according to
the affirmation or negation of the dictum. So that . . . this
modal 'that Socrates runs is possible' is affirmative, while
'that Socrates runs is not possible' is negative.
There are two notable points in this text. First there is the very
thorough formalism, the modal proposition being classified accord-
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SCHOLASTIC LOGIC
ing to the place which the mode has in it. Then there is the explicit
distinction of the two structures, one of which Aristotle made the
basis of his modal logic (§ 15, B), the other of which Theophrastus
adopted (§ 17, B). The modals de re correspond to the Aristotelian
structure, in which the mode does not determine the 'composition'
or, as we should say, the proposition as a whole, but 'the predicate'.
Taking the proposition A is possibly BJ as de re, we could analyze
it thus:
if x is A, then x is possibly B.
But the modals de dido have the Theophrastan structure, according
to which the fore-going proposition will be taken as de dido and can
be interpreted:
that A is B is possible.
D. COMPOSITE AND DIVIDED SENSES
Closely connected with that doctrine, classical in Scholasticism,
is that of the composite and divided senses of propositions. It was
developed out of the Aristotelian theory of the fallacies of division
and composition (11.22f.), and partly corresponds to the foregoing
analysis of modal propositions (29.09), but extends to other kinds
as well. It seems to have secured a quite central place in later
scholastic logic. We cite first a text of Peter of Spain:
29.10 There are two kinds of composition. The first kind
arises from the fact that some dictum can suppose for itself
or a part of itself, e.g. 'that he who is sitting walks is possible'.
For if the dictum 'that he who is sitting walks' is wholly
subjected to the predicate 'possible', then the proposition is
false and composite, for then opposed activities, sitting and
walking, are included in the subject, and the sense is: 'he
who is sitting is walking'. But if the dictum supposes for a part
of the dictum, then the proposition is true and divided, and
the sense is: 'he who is sitting has the power of walking'. To
be distinguished in the same fashion is : 'that he who is not
writing is writing is impossible'. For this dictum 'that he
who is not writing is writing' is subjected to the predicate
'impossible',* but sometimes as a whole, sometimes in respect
of a part of itself. And similarly: 'that a white thing is black
is possible'. And it is to be known that expressions of this
kind are commonly said to be de re or de dicio.
Reading impossibile for possibile.
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PROPOSITIONS
A twofold terminology can be seen here ; the distinction composita-
divisa corresponds to de dido - de re. Peter also introduces the notion
of supposition, while Thomas (29.09) proceeds wholly syntactically.
Thomas has yet other expressions for the same idea:
29.11 Further (it is objected), if everything is known by
God as seen in the present, it will be necessary that what God
knows, is, as it is necessary that Socrates sits given that he is
seen to be sitting. But this is not necessary absolutely, or as is
said by some, by necessity of the consequent: rather condi-
tionally, or by necessity of consequence. For this conditional
is necessary: If he is seen to be sitting, he sits. Whence also, if
the conditional is turned into a categorical, so that it is said:
what is seen to be sitting, necessarily sits, evidently if this is
understood as de dido and composite, it is true ; but understood
as de re and divided, it is false. And so in these and all similar
cases . . . people are deceived in respect of composition and
division.
This gives us the two following series of expressions, correspond-
ing member to member (the word propositio being understood with
each) : de dido, composita, necessaria necessitate consequentiae,
necessaria sub conditione - de re, divisa, necessaria necessitate conse-
queniis, necessaria absolute.
Paul of Venice gives a peculiar variant of the doctrine of de dido
and de re:
29.12 Some say that always when the mode simply precedes
or follows the expression with the infinitive, then the sense is
definitely called 'composite' in every case, e.g. 'it is possible
that Socrates runs', 'that Socrates runs (Socratem currere) is
possible'. But when the mode occupies a place in the middle the
sense is called 'divided', e.g. 'for Socrates it is possible to
run'. Others on the other hand say that when the mode
simply precedes, the sense is composite, as previously, but
when it occupies a middle place or comes at the end, then the
sense is divided, e.g. 'of A I know that it is true', 'that A is
true is known by me'. And so with others similar.
But though these ways of speaking enjoy probability, yet
they are not wholly true. ... So I say otherwise, taking a
position intermediate between them: when the mode simply
precedes a categorical or hypothetical dictum, it effects the
composite sense; and when it occurs between the verb and
the first extreme, it is taken in the divided sense; but when it
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SCHOLASTIC LOGIC
follows at the end, it can be taken in the composite or the
divided sense.
This seemingly purely grammatical text is yet not without
interest as showing how scholastic logic at the end of the 14th
century was wholly bent on grasping the laws of everyday language.
We find no essentially new range of problems in Paul beyond those
of Thomas and Peter.
There is yet another interpretation of the composite and divided
senses, first found in Peter, in a text which seems to adumbrate all
the associated problems:
29.13 (The fallacy of) division is a false division of things
that should be compounded. There are two kinds of division.
The first arises from the fact that a conjunction can conjoin
either terms or propositions, e.g. . . . Tive is even or odd'.
Similarly: 'every animal is rational or irrational'. For if this
conjunctive particle 'or' divides one proposition from another,
it is false, and its sense is: 'every animal is rational or every
animal is irrational'. If it disjoins one term from another,
then it is true and its sense is: 'every animal is rational or
irrational', in which the whole disjunctive complex is predi-
cated. Similarly: 'every animal is healthy or sick', 'every
number is even or odd'.
A more exact formulation of the same thought occurs in Burleigh :
29.14 'Every animal is rational or irrational.' The proof is
inductive. The disproof runs: every animal is rational or
irrational, but not every animal is rational, therefore every
animal is irrational. The conclusion is false, the minor is not
(false), therefore the major is. The consequence is evident
from the locus of opposites (16.18).
Solution. The first (proposition) is multiple, according to
composition and division. In the sense of composition it is
true, in the sense of division it is false. Induction does not
hold in the sense of division, because in the sense of division
there is not a categorical proposition but a hypothetical of
universal quantity; and thus the answer to the proof is
clear.
To the disproof, I say the consequence does not hold in the
sense of composition, nor is there room for an argument from
the locus of opposites, for the locus of opposites is when one
argues from a disjunctive and the negation of one part, to
186
PROPOSITIONS
the other part; but in the sense of composition this is not a
disjunctive but a categorical proposition.
The form of propositions of this kind in the composite sense could
be expressed with variables thus:
(1) for every x: x is A or a; is B.
In the divided sense the same proposition could be interpreted :
(2) Tor every x: x is A, or for every x: x is B.
If this interpretation is correct we have here an important theorem
about the distribution of quantifiers. Yet Burleigh does Dot seem
to have been thinking quite of (1), but rather of
(V) for every x: x is (A or B).
E. REFERENCE OF PROPOSITIONS
Finally we give a text in which the chief scholastic theories about
the semantic correlate of propositions are listed. It is taken from
Paul of Venice.
29.15 About the essence of the proposition . . . there are
many opinions.
The first is that the significate of a true proposition is a
circumstance (modus) of the thing and not the thing itself. . . .
29.16 The second opinion is that the significate of a true
proposition is a composition of the mind (mentis) or of the
intellect which compounds or divides. . . .
29.17 The third opinion, commonly received among the
doctors of my (Augustinian) Order, in particular by Master
Gregory of Rimini, is that the significate of a proposition is
whatever in any way exists as a signifiable complex. And
when it is asked whether such a signifiable is something or
nothing, he answers that the name 'something' and its syno-
nyms 'thing' and 'being' can be understood in three ways.
(1) First in the widest sense, according to which everything
signifiable, with or without complexity, truly or falsely, is
called 'thing ' and 'something'. ... (2) In a second way these
(names) are taken for whatever is signifiable, with or without
complexity, but truly, ... (3) In a third way the aforesaid
names are taken in such wise that they signify some existent
essence or entity, and in this way, what does not exist is
called 'nothing' .... So this opinion says that the significate
of a proposition is something, if one takes the afore-mentioned
terms in the first or second way. . . .
29.18 The fourth opinion posits some theses. (1) The first
is this : that no thing is the adequate or total significate of a
187
SCHOLASTIC LOGIC
mental proposition properly so called; since every such
(proposition) signifies a variety of mutually distinct things,
by reason of its parts to which it is equivalent in its signifying.
And this is evident to everyone who examines the matter.
Hence there is no total or adequate significate of such a
proposition.
(2) The second thesis: whatever is signified by a mental
proposition properly so called according to its total significa-
tion is also signified by any of its parts. . . .
(3) Third thesis: no dictum corresponding to a mental
proposition properly so called, e.g. an expression in the
infinitive mood taken as significant, supposes for any thing.
For instance, if the dictum, i.e. the expression in the infinitive
mood, 'that man is an animal', corresponding to the proposi-
tion 'man is an animal', is taken materially it stands for some
thing, namely for the proposition to which it corresponds;
but if it be taken significatively, i.e. personally, then according
to the fourth opinion it stands for no thing. This is evident,
since such an expression, so taken, signifies a number of
things, viz. all those signified by the corresponding proposition,
and so there would be no reason for it to suppose for one of
its significates rather than another; hence (it supposes)
either for each or for none. But nobody would say for each,
since the expression 'that man is an animal' would signify
an ass or suppose for an ass. Therefore for none. And what is
said of that instance, holds for any other.
The four opinions there enumerated could be summed up thus
in modern terminology; a proposition has for its semantical correlate
(1) a real fact, (2) a psychical act ( 3) an objective content (the Stoic
lekton), (4) nothing at all beyond what its parts already signify.
In the 15th century there were very complicated and sharp disputes
about this problem. But as it lies on the border-line of pure logic
we shall omit consideration of them here (29.19).
188
II. PROPOSITIONAL LOGIC
§ 30. NOTION AND DIVISION OF CONSEQUENCES
A. HISTORICAL SURVEY
The theory of consequences is one of the most interesting scholastic
doctrines. Essentially it is a development of Stoic prepositional
logic, though so far as is known it was constructed entirely anew,
not in connection with the Stoic logoi (§ 21 ) but with certain passages
of the Hermeneia and, above all, the Topics. All the same, fragments
of the Stoic propositional logic did influence the Scholastics, mostly
through the mediation of Boethius, though for a few we must
suppose some other sources, as e.g. for the 'dog-syllogism' (22.19;,
which is found in Thomas Aquinas (30.01). But that these fragments
were not the starting point is clear from the fact that, at least to
begin with, they are not cited in the treatise on consequences, but
in another on hypothetical syllogisms.
The name ' consequential' is Boethius's translation of Aristotle's
axoXoi>07](7!.<; which occurs frequently in the Hermeneia (30.02) but
not in any exact technical sense, rather for following quite in general.
The word has the same sense in Abelard, though limited to logical
relationships between terms (30.03),* and to some extent also in
Kilwardby * * (30.04) and Peter of Spain (30.05) E.g. in the latter we
read of a consequenlia esseniiae (30.06).
In Ockham, on the other hand, and his successors the word
has a sharply defined technical meaning, and signifies a relation of
consequence between two propositions.
The following text from Kilwardby may serve as a good example
of the earlier stage:
30.04 He (Aristotle) also says that something is a consequent
(of something else) in part, and yet whatever follows from A
follows from all that is contained under A, since what follows
on the consequent follows on the antecedent, and so every
consequent follows on the whole antecedent. . . .
It is to be answered to this, that (Aristotle) in this whole
treatment (Prior Analytics) takes 'consequent' for the pre-
dicate and 'antecedent' for the subject. . . .
A further passage from Kilwardby is extremely instructive about
this relationship:
Vide Translator's Preface, B.
Cited from transcriptions of two Oxford MSS (30.04) made by the translator.
189
SCHOLASTIC LOGIC
30.07 Consequence is twofold, viz. essential or natural,
as when a consequent is naturally understood in its antecedent,
and accidental consequence. Of the latter kind is the conse-
quence according to which we say that the necessary follows
on anything. . . .
That shows that for Kilwardby a 'natural' and 'essential' conse-
quence is only present when it is a matter of connection between
terms. Thus for him the proposition 'every man runs, therefore
there is a man who runs' would be natural, since 'each man' is
'naturally' included in 'every man'. In other words it is for him
always based on term-logical relationships. He does also recognize
purely propositional consequences such as the one he states: 'the
necessary follows on anything', but these he considers only 'acciden-
tal' and of an inferior kind.
This opinion of Kilwardby's is of interest as showing that the
Scholastics did not take the abstract propositional logic of the Stoics
as their starting point, but the term-logic of Aristotle. Yet before
very long they built on that basis a technically excellent pure logic
of propositions that to the best of our knowledge was superior to
that of the Megarians and Stoics.
Since the paper of Lukasiewicz (30.08) more works have been
devoted to this propositional logic than to any other scholastic
logical doctrine, and it is better known than most others (30.09). Yet
we are still far from having a complete knowledge of it. We cannot
here enumerate all the scholastic consequences that have been
investigated in the 20th century, but must limit ourselves to texts
defining the notion of consequentia, and then (§ 31) give a few examp-
les.
B. DEFINITION OF CONSEQUENCE
Pseudo-Scotus gives the following definition of consequence:
30.10 A consequence is a hypothetical proposition composed
of an antecedent and consequent by means of a conditional
connective or one expressing a reason (rationalis) which
signifies that if they, viz. the antecedent and consequent,
are formed simultaneously, it is impossible that the ante-
cedent be true and the consequent false.
Here a consequence is conceived as a proposition in almost word
for word agreement with the Stoic definition (19.15) with only two
considerable differences: (1) 'proposilio' means, not the lekton, but
the thought, written and spoken proposition (cf. 26.03) ; (2) the
consequence corresponds to the compound and inferential sentences
of the Stoics (19.15). Implication is defined in the Diodorean way
190
CONSEQUENCES
(20.08), though the time-variables might be thought to be missing;
but comparison with the definition of consequence ut nunc (30.12,
cf. 30.16) shows that Diodorus's idea of 'for all times' is basic for
the Scholastics too. They conducted a complicated discussion which
shows that the range of problems considered was much wider than
might be expected from what we have said here (cf. 30.17 f.).
A noteworthy exception to the premise that a consequence is a
proposition* is found in Burleigh:
30.11 It is also to be noted that the (contradictory) opposite
of the antecedent does not follow from the opposite of the
consequent in every valid consequence, but only in non-
syllogistic consequences. For in syllogistic consequences the
antecedent has no opposite, because a syllogistic antecedent
is an unconnected plurality of propositions (propositio plures
inconiuncle) and because such an antecedent has no opposite
at all, it not being a proposition that is either simply or
conjunctively one. But in a syllogistic consequence the
opposite of one premiss follows from the opposite of the
conclusion with the other premiss. And if from the opposite of
the conclusion with one or other of the premisses there
follows the opposite of the remaining premiss, then the original
syllogism was valid. For that is how the Philosopher proves
his syllogisms, viz. arguing from the opposite of the con-
clusion with one of the premisses, as can be seen in the first
book of the Prior Analytics.
'Propositio plures inconiuncle' means here, as usually among the
Scholastics (cf. 35.45) not a compound proposition, not^therefore a
product of propositions, but a number of juxtaposed propositions.
It follows that syllogisms, and so 'syllogistic consequences' are
not propositions, and further that consequences were not always
thought of as conditional propositions.
C. DIVISION OF CONSEQUENCES
Here again we begin with Pseudo-Scotus :
30.12 Consequences are divided thus: some are material,
others are formal. A formal consequence is one which holds
in all terms, given similar mutual arrangement (disposilio)
and form of the terms. ... A material consequence is one
* Prof. L. Minio-Paluello tells us in connection with Cod. Orleans '266, fol. 78
that this was already debated in the middle of the 12th century.
191
SCHOLASTIC LOGIC
which does not hold in all terms given similar mutual arrange-
ment and form so that the only variation is in the terms
themselves. And such a consequence is twofold : one is simply
true, the other true for the present (ut nunc). A simply true
consequence is one reducible to a formal consequence by the
addition of a necessary proposition. A correct material
consequence true for the present is one which is reducible to
a formal consequence by the addition of a true contingent
proposition.
So there are three kinds of consequence: (1) formal, (2) simple
material, (3) material ut nunc. The last two are reduced to the first,
but by means of different kinds of proposition. For (2) there is requir-
ed a necessary, and so always true, proposition, for (3) one must
use a contingent proposition, one which is therefore true only at a
certain time. An example of the reduction of (2) to (1) is: 'A man
runs, therefore an animal runs' is reduced to a consequence of kind
(1) by means of the proposition 'every man is an animal' when it is
said: 'every man is an animal, a man runs, therefore an animal
runs'. The newly introduced proposition is necessary, and so always
true, hence the consequence reduced by its means to (1) is 'simply'
valid, valid for all time.
Another definition is to be found in Ockham, along with a further
division of formal consequence:
30.13 Of consequences, one kind is formal, another material.
Formal consequence is twofold, since one holds by an extrinsic
medium concerning the form of the proposition, such as
these rules: 'from an exclusive to a universal (proposition)
with the terms interchanged is a correct consequence', 'from
a necessary major and an assertoric minor (premiss) there
follows a necessary (conclusion)' etc. The other kind holds
directly through an intrinsic medium and indirectly through
an extrinsic one concerning the general conditions of the
proposition, not its truth, falsity, necessity or impossibility.
Of this kind is the following: 'Socrates does not run, therefore
some man does not run'. The consequence is called 'material'
since it holds precisely in virtue of the terms, not in virtue
of some extrinsic medium concerning the general conditions
of the proposition. Such are the following: 'If a man runs,
God exists', 'man is an ass, therefore God does not exist' etc.
This text is most important, since Ockham here introduces a
doctrine analogous to that of Whitehead and Russell in their distinc-
tion of formal and material implication (44.11 ff.), analogous only,
192
CONSEQUENCES
because the basic idea of implication is here Diodorean (20.08)
instead of Philonian (20.07). Formal implications in this sense are
further divided into two classes according as they hold in virtue of
their component symbols or other propositions of the system.
These ideas are defined with some accuracy in a text of Albert of
Saxony:
30.14 Of consequences, one kind is formal, another
material. That is said to be a formal consequence to which
every proposition which, if it were to be formed, would be a
valid consequence, is similar in form, e.g. lb is a, therefore
some a is b\ But a material consequence is one such that not
every proposition similar in form to it is a valid consequence,
or, as is commonly said, which does not hold in all terms when
the form is kept the same; e.g. 'a man runs, therefore an
animal runs'. But in these (other) terms the consequence is
not valid: 'a man runs, therefore a log runs'.
We may compare 26.11 f. with this text, so far as concerns the
notion of logical form, and indeed the former follows immediately
on the latter.
Reverting to the distinction of simple and ut nunc consequences
(30.12) with a similar reference to time-variables as in Diodorus
Cronus (20.08, cf. 19.23), Burleigh formulates this idea explicitly
and accurately:
30.15 Of consequences, some are simple, some ut nunc.
Simple are those which hold for every time, as: 'a man runs,
therefore an animal runs'. Consequences ut nunc hold for a
determinate time and not always, as: 'every man runs,
therefore Socrates runs' ; for that consequence does not hold
always, but only so long as there is a man Socrates.
The first rule of consequence is this : in every valid simple
consequence the antecedent cannot be true without the con-
sequent. And so, if in any possible given case the antecedent
could be true without the consequent, the consequence
would not be valid. But in a consequence ut nunc, the ante-
cedent ut nunc, i.e. for the (given) time for which the conse-
quence holds, cannot be true without the consequent.
Buridan has a text on this subject, in which occurs the new idea
of consequence for such and such a time (ut tunc).
30.16 Of material consequences some are said to be
consequences simply, since they are consequences without
qualification, it being impossible for their antecedents to be
true without their consequents. . . . And others are said to be
193
SCHOLASTIC LOGIC
consequences ut nunc, since they are not valid without
qualification, it being possible for their antecedents to be true
without their consequents. However, they are valid ut nunc,
since things being exactly the same as they now are, it is
impossible for the antecedent to be true without the conse-
quent. And people often use these consequences in ordinary
language (utuntur saepe vulgares), as when we say: 'the white
Cardinal has been elected Pope' and conclude: 'therefore a
Master in Theology has been elected Pope' ; and as when I
say: 'I see such and such a man' . . . you conclude 'therefore
you certainly see a false man'. But this consequence is reduced
to a formal one by the addition of a true, but not necessary,
proposition, or of several true and not necessary ones, as in
the examples given, since the white Cardinal is a Master in
Theology and since such and such a man is a false man. In
that way the following is a valid consequence: under the
hypothesis that there are no men but Socrates, Plato and
Robert, 'Socrates runs, Plato runs and Robert runs; therefore
every man runs'. For this consequence is perfected by this
true (proposition): 'every man is Socrates, Plato or Robert'.
And it is to be known that to this kind of consequences ut
nunc belong permissive consequences, e.g. 'Plato says to
Socrates: if you come to me I will give you a horse'. The
proposition may be a genuine consequence, or it may be a
false proposition and no consequence, since ( 1 ) if the antecedent
is impossible, viz. because Socrates cannot come to Plato,
then the consequence is simply speaking a genuine conse-
quence, because from the impossible anything follows as will
be said below. But if (2) the antecedent is false but not
impossible, then the consequence is valid ut nunc, because
from whatever is false anything follows, as will be said later,
provided, however, that we restrict the name 'consequence
ut nunc' to consequences ut tunc, whether concerning the
past, future, or any other determinate time. But if the ante-
cedent is true, so that Socrates will come to Plato, then
perhaps we should say that it is still a genuine consequence
because it can be made formal by the apposition* of true
(propositions), when one knows whatever Plato wills to do in
the future, that his wish will persist and that he will be able
to carry it out; and when all circumstances are taken account
* Reading appositas for oppositas.
194
CONSEQUENCES
of according to which he wills it, and he suffers no hindrance,
so that he will be able to and will do what and when he wills;
if you then modify this proposition so that it is true according
to the ninth book of the Metaphysics, i.e. 'Plato wills to give
Socrates a horse when he comes to him; therefore Plato will
give Socrates a horse'. If then these propositions about Plato's
will and power are true, then Plato uttered a genuine conse-
quence ui nunc to Socrates, but if they are not true he told
Socrates a lie.
D. MEANING OF IMPLICATION
If the rooks and the crows cawed about the meaning of implication
in the 2nd century b. c. (20.06), this occupation was surely intensi-
fied in the 15th century. For while the Megarian-Stoic school has
bequeathed to us only four interpretations, Paul of Venice tells us
of ten. Not all his definitions are comprehensible to us today, but
perhaps this is due to textual corruptions. However, for the sake of
completeness we give the whole list.
30.17 Some have said that for the truth of a conditional
is required that the antecedent cannot be true without the
consequent. . . .
Others have said that for the truth of a conditional it is
not required that the antecedent cannot be true without
the consequent in the divided sense, but it is required that it
is not possible for the antecedent to be true without the
consequent being true.
Thirdly people have said that for the truth of a conditional
it is required that it is not possible that the antecedent of
that consequence be true unless the consequent be true . . .
Fourthly people have said that for the truth of a conditional
it is required that it is not possible that the antecedent be
true while the consequent of that same antecedent is false
without a fresh interpretation (impositio) ....
Fifthly people say that for the truth of a conditional it is
required that if things are (ita est) as is signifiable by the
antecedent, necessarily things are as is signifiable by the
consequent. . . .
Sixthly people say that for the truth of a conditional it is
required that it be not possible that things should be so and
not so, referring to the significates of the antecedent and of
the consequent* of that conditional. . . .
* Omitting oppositi.
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SCHOLASTIC LOGIC
Seventhly people say that for the truth of a conditional it
is required that it is not possible for things to be so and not so,
referring to the adequate significates of the antecedent and
the consequent. . . .
Eighthly people say that for the truth of a conditional it is
required that the consequent be understood in the antece-
dent. . . .
Ninthly people say that for the truth of a condition it is
required that the adequate significate of the consequent be
understood in the antecedent.
Tenthly people say that for the truth of a conditional it is
required that the opposite of the consequent be incompatible
with the antecedent. . . .
For the distinction between the first two of those definitions the
following text of Buridan is instructive. *
30.18 Then there is the rule . . . , that the consequence is
valid when it is impossible that things are as signified by the
antecedent without their being as is signified by the conse-
quent. And this rule can be understood in two ways.
In one way so that it would be a proposition concerning
impossibility in the composite sense (the way in which it is
usually intended) and the meaning then is that a consequence
is valid when the following is impossible: 'If it is formed,
things are as is signified by the antecedent, and are not as
is signified by the consequent.' But this rule is invalid, since
it justifies the fallacy: 'No proposition is negative, therefore
some propositions are negative.'
In the other way, so that it would be a proposition concern-
ing impossibility in the divided sense, so that the meaning is:
a consequence is valid when whatever is stated in the ante-
cedent cannot possibly be so without whatever is stated
in the consequent being so. And it is clear that this rule would
not prove the fallacy true; for whatever 'no proposition is
negative' states, is possibly so, although things are not as the
other (proposition of the fallacy) states; for if they were,
affirmatives would persist but all negatives would be annihi-
lated.
* This text was kindly communicated by Prof. E. Moody.
196
CONSEQUENCES
E. DISJUNCTION
The notion of implication is closely connected with that of dis-
junction. Hence two characteristic texts are in place to illustrate
the problems connected with the latter. Peter of Spain writes:
30.19 For the truth of a disjunctive (proposition) it is
required that one part be true, as 'man is an animal or crow is
a stone', and it is allowed that both of its parts be true, but
not so properly, as 'man is an animal or horses can whinny'.
For its falsity it is required that both of its parts be false, as
'man is not an animal or horse is a stone'.
Peter's idea of disjunction is evidently rather hazy, for he wavers
between the exclusive (20.14) and the non-exclusive (20.18) dis-
junction, describing the latter as 'less proper' though at the same
time determining falsity in a way suitable to it alone. Which of the
two is 'proper' must have been debated even in the 14th century,
as can be seen from Burleigh's fine text:
30.20 Some say that for the truth of a disjunctive it is
always required that one part be false, because if both parts
were false it would not be a true disjunctive; for disjunction
does not allow those things which it disjoins to be together,
as Boethius says. But I do not like that. Indeed I say that
if both parts of a disjunctive are true, the whole disjunctive
is true. And I prove it thus. If both parts of a disjunctive are
true, one part is true; and if one part is true, the disjunctive
is true. Therefore (arguing) from the first to the last: if both
parts of a disjunctive are true, the disjunctive is true.
Further, a disjunctive follows from each of its parts, but it
is an infallible rule that if the antecedent is true, the conse-
quent is true; therefore if each part is true the disjunctive is
true.
I say therefore, that for the truth of a disjunctive it is not
required that one part be false.
Burleigh therefore definitely sides with those who understand
disjunction as non-exclusive. Also to be remarked in this text are
the two propositional consequences formulated with exemplary
exactness:
30.201 If A and B, then A.
30.202 If A, then A or B.
197
§31. PROPOSITIONAL CONSEQUENCES
The Scholastics made no explicit difference between conse-
quences pertaining to propositional and to term-logic. Yet they
usually, at least after Ockham, dealt with the former first. It is
convenient in this connection to quote a text from Paul of Venice in
which he collects the terminology used of so-called hypothetical
propositions. After that we give three series of texts, one from
Kilwardby (first half of the 13th century), one from Albert of Saxony
(second half of the 13th century) and the third from Paul of Venice
(first half of the 15th century). To those we add some texts from
Buridan about consequences ut nunc. We cannot claim to survey
even the essentials of scholastic propositional logic, for this is as
yet too little explored. The texts cited serve only as examples of
the problems considered and the methods applied.
A. HYPOTHETICAL PROPOSITIONS
31.01 Some posit five kinds of hypotheticals, some six,
others seven, others ten, others fourteen etc. But leaving all
those aside, I say that there are three and no more kinds of
hypotheticals that do not coincide in significance, viz. the
copulative, disjunctive, and conditional to which the rational
is to be counted equivalent. For I do not see that the temporal,
local and causal are hypothetical, still less those formed and
constituted by other adverbial and connective particles. These
are only hypothetical by similitude, e.g. 'I have written as you
wanted', 'Michael answers as I tell him'. Similarly the com-
parative, e.g. 'Socrates is as good as Plato', 'Socrates is whiter
than Plato'. Again, the relative, e.g. 'I see a man such as
you see'. . . . Similarly the inhibitive, e.g. 'Socrates takes
care than no-one confute him'. Again the elective, e.g. 'it is
better to concede that your reply is bad than to concede
something worse'. Similarly the subjunctive, e.g. 'I saw to it
that you answered well'. Similarly the expletive, e.g. 'you
may be moving but you are not running!' Thus by taking*
the other particles in turn one can form a very great number
of (pretended) hypotheticals.
B. KILWARDBY
We take a first series of consequences from Kilwardby's com-
mentary on the Prior Analytics of Aristotle. Kilwardby does not
* Reading discurrendo for distribuendo.
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CONSEQUENCES
always distinguish very clearly between propositional and term-
relationships (cf. 30.04), so that 'antecedent' and 'consequent' must
sometimes be understood as referring to the subject and predicate
of universal affirmative propositions.
31.02 What is understood in some thing or things, follows
from it or from them by a necessary and natural consequence ;
and so of necessity if one of a pair of opposites is repugnant to
the premisses (of a syllogism) the other follows from them.
31.03 If one of the opposites does not follow, the other
can stand.
31.04 If one of the opposites stands, the other cannot.
31.05 What does not follow from the antecedent does
not follow from the consequent.
31.06 What follows from the consequent follows from the
antecedent.
31.07 What is compossible with one of two equivalents
(convertib ilium) is so with the other.
31.08 It is to be said that a negation can be negated, and
so there is a negation of negation, but this second negation
is really an affirmation, though accidentally (secundum quid)
and vocally a negation. For a negation which supervenes on
a negation destroys it, and in destroying it posits an affirma-
tion.
31.09 If there necessarily follows from 'A is white' lB is
large', then from the denial (destrudio) of the consequent: if
B is not large, A is not white.
31.10 A disjunctive follows from each of its parts, and by a
natural consequence; for it follows: if you sit, then you sit
or you do not sit.
31.11 If the antecedent is contingent or possible, so is the
consequent.
31.12 It is not necessary that what follows from the ante-
cedent follows from the consequent.
C. ALBERT OF SAXONY
Secondly we give a series of texts from the Peruiilis Logica of
Albert of Saxony in which the doctrine of consequences can be seen
in a highly developed state. Albert is here so closely dependent on
Buridan that he often simply copies him. But there is much that he
formulates more clearly, and the available text of Buridan is not so
good as that of the Peruiilis Logica. Buridan himself is not the
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SCHOLASTIC LOGIC
original author of his doctrine of consequences; much of it comes
from Ockham, and some even from Peter of Spain.
As in this whole section, the contemporary range of problems is
only barely illustrated.
Albert's definitions of antecedent and consequent deserve to be
quoted first:
31.13 That proposition is said to be antecedent to another
which is so related to it that it is impossible that things be
as is signifiable by it without their being as is in any way
signifiable by the other, keeping fixed the use (impositio) of
the terms.
Like all Scholastics of the 14th century and after, Albert makes a
clear distinction between a rule of consequence and the consequence
itself. A rule is a metalogical (more exactly a meta-metalogical)
description of the form of a valid consequence. The consequence
itself is a proposition having this form. That generally holds good;
but some of Albert's rules are conceived as propositional forms like
the Stoic inference-schemata (21.22) - cf. the fifth (31.18) - only
with this difference, that the variables are here evidently metalo-
gical, i.e. to be substituted with names of propositions, not with
propositions themselves as is the case with the Stoic formulae.
31.14 The first (rule of simple consequence) is this: from an
impossible proposition every other follows. Proof: from the
nominal definitions of antecedent and consequent given in the
first chapter. For if a proposition is impossible, it is impossible
that things are as it indicates, and are not as any other
indicates ; therefore the impossible proposition is antecedent to
every other proposition, and hence every proposition follows
from an impossible one. This it is which is usually expressed:
anything follows from the impossible. And so it follows: man
is an ass, therefore a man runs; since the antecedent being
impossible, if things are not as the consequent indicates, it is
impossible that they should be as the antecedent indi-
cates.
31.15 Second rule: A necessary proposition follows from
any proposition. This is again proved by the nominal defini-
tions of antecedent and consequent. For it is impossible that
things should not be as a necessary proposition indicates,
if they are as any other (proposition) indicates. Hence a
necessary proposition is a consequent of any proposition. It
follows therefore that this consequence is valid: 'a man runs,
therefore God exists', or '(therefore) ass is an animal'.
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CONSEQUENCES
The proofs of these two rules are very typical of the Scholastic
approach to propositional logic and show how different it is both to
that of the Megarian-Stoics and that of the moderns. The essential
scholastic point is that a consequence does not unite two states of
affairs but two propositions (in the scholastic sense, which includes
the mental propositions, cf. 26.03). Let 'P' be the name of the
proposition expressing the state of affairs p, and lQ' the name of
the proposition expressing the state of affairs q, the proof of the
first consequence can be presented thus:
As axiom is presupposed
31.151 If p cannot be the case then (p and q) cannot be the
case.
Then the process is:
(1) P is impossible (hypothesis)
(2) p cannot be the case (by (1) and the definition of
impossibility)
(3) (p and not q) cannot be the case (by (2) and 31.151)
(4) Q is the consequent of P (by the definition and (3))
(5) Q follows from P (by definition)
And this was to be proved, Q being any proposition.
Thus we can see that a metalogical thesis about a relationship
(consequence) between propositions is proved through reduction to
logical laws concerning relationships between states of affairs.
31.16 Third rule: (1) From any proposition there follows
every other whose contradictory opposite is incompatible
with it (the first). And (2) from no proposition does there
follow another whose contradictory opposite is compatible
with it, where (the expression) 'a proposition is compatible
with another' is to be understood in the sense that the state of
affairs (sic esse) which the one indicates is compatible with that
which the other indicates. . . .
The first part of the rule is proved (thus) : Let us suppose
that the proposition B is incompatible with the proposition A.
I say (then) that from A there follows the contradictory
opposite of B, i.e. not-B. This is evident, for A and B are
incompatible and therefore (either A) is impossible, so that
every proposition follows from it, by the first rule; or A is
possible, then necessarily if A is the case, either B or not-B is
the case, since one part of a pair of contradictory opposites is
always true. But it is impossible that if A is the case, B is the
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SCHOLASTIC LOGIC
case, by hypothesis. Therefore it is necessary that if A is the
case, not-B is the case. Therefore not-B follows from A.
The second part of the rule is proved (thus) : if A and
not-B are true together, then this holds: If A is the case,
B is not the case. But since B and not-B are not true together,
it is possible that if A is the case, B is not the case. Therefore
B does not follow from A.
31.17 Fourth rule: for every valid consequence, from the
contradictory opposite of the consequent there follows the
contradictory opposite of the antecedent. This is evident,
since on the supposition that B follows from A, I say that
not-^4 follows from not-B. For either it is so, or it is possible
that A and not-B are true together, by the previous rule. But
it is necessary that if A is the case, B is the case. Therefore B
and not-B will be true together, which is impossible, by the
accepted (communis) principle 'it is impossible that two contra-
dictories should be true together'. . . .
31.18 Fifth rule: if B follows from A, and C from B, then (1)
C follows from A ; and (2) C follows from everything from
which B follows; and (3) what does not follow from A, does
not follow from B; and (4) from everything from which C does
not follow, B too does not follow. That is to say, in current
terms, all the (following) consequences are valid: (1) Whatever
follows from the consequent follows from the antecedent;
(2) The consequent of this consequence follows from all that
from which the antecedent follows; (3) What does not follow
from the antecedent does not follow from the consequent;
(4) The antecedent does not follow from that from which the
consequent does not follow. This rule has four parts.
The first (part) is: If B follows from A, and C from B, then
C follows from A. For on the supposition that B follows from
A, if things are as A indicates, they are also as B indicates, by
the nominal definition of antecedent and consequent. And
on the supposition that C follows from B, if things are as B
indicates, they are also as C indicates. Therefore, if things
are as A indicates, they are also as C indicates. And accordingly
C follows from A.
The second part is evident, since nothing from which B
follows can be the case if B is not the case ; and as B cannot be
the case if C is not the case, it follows also: C follows from all
from which B follows. And by 'being the case' is to be under-
stood : being as B indicates ....
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CONSEQUENCES
31.19 Sixth rule: (1) It is impossible that false follows from
true. (2) It is also impossible that from possible follows
impossible. (3) It is also impossible that a not necessary
proposition follows from a necessary one. (The first part)
is evident by the nominal definition of antecedent and conse-
quent. For if things are as the antecedent indicates, they are
also as the consequent indicates, and accordingly, when the
antecedent is true, the consequent is true and not false. The
second part is evident, for if things can be as the antecedent
indicates, they can also be as the consequent indicates; and
accordingly, when the antecedent is possible, the consequent
also (is possible). The third part is evident, for if things
necessarily are as the antecedent indicates, they must also
(necessarily) be as the consequent indicates.
31.20 There follows from this rule: (1) if the consequent
of a consequence is false, its antecedent is also false ; (2) further,
if the consequent of a consequence is impossible, its antece-
dent is also impossible ; (3) further if the consequent of a con-
sequence is not necessary, its antecedent also is not necessary.
And I purposely (notanter) say, 'if the consequent is not
possible' and not 'if the consequent is not possibly true', since
in this (consequence) : 'every proposition is affirmative,
therefore no proposition is negative', its antecedent is possible
and its consequent too is possible, but although it is possible, it
is impossible that it be true, as was said above. And yet true
can follow from false, and possible can follow from impossible,
and necessary can follow from not necessary, as is evident
from Aristotle in the second (book) of the Prior (Analytics,
ch. 2) (16.32)
31.21 Seventh rule: if B follows from A together with one
or more additional necessary propositions, then B follows
from A alone. Proof: B is either necessary or not necessary. If
it is necessary, it follows from A alone, by the second rule,
since the necessary follows from any (proposition). But if B
is not necessary, then A is either possible or impossible.
Suppose A is impossible then again B follows from A alone as
also from A with an additional necessary proposition, by the
first rule. Since from the impossible, anything follows. But
suppose A is possible, then if A is the case it is impossible that
B is not the case, or, if A is the case it is possible that B is
not the case. On the first supposition, B follows from A alone,
as also from A with an additional necessary proposition, by the
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SCHOLASTIC LOGIC
nominal definition of antecedent and consequent. But sup-
posing that if A is the case it is possible that B is not the case,
then if A is the case, A and the additional necessary pro-
position must be true together. For it is impossible that* A
should not be the case, since it is not possible that if A is the
case, A is not the case. And accordingly, granted that A is the
case, i.e. granted that things are as A indicates, it is necessary
that they should be as A and the additional necessary pro-
position indicate. Therefore from A there follow A and the
additional necessary proposition. And as B follows from A
and the additional necessary proposition, one obtains the
probandum by means of the first part of this rule, (viz.)
that B follows from A alone, which was to be proved.
The rule could be formulated:
31.211 If C is necessary, then : if B follows from A and C, B
follows from A alone,
and the proof is contained in the words: 'if A is the case, A and the
additional necessary proposition must be the case' and the subse-
quent justification. For in fact, if C is necessary, then if we have A,
we have A and C, and then if B follows from A and C, B follows
from A. The passage previous to the words just quoted is therefore
superfluous, but it has been retained as characteristic of the scho-
lastic approach.
31.22 Eighth rule: every consequence of this kind is for-
mal: 'Socrates exists, and Socrates does not exist, therefore a
stick stands in the corner'. Proof: By formal consequence it
follows: 'Socrates exists and Socrates does not exist, therefore
Socrates exists', from a complete copulative proposition to
one of its parts. Further it follows: 'Socrates exists and
Socrates does not exist, therefore Socrates does not exist' by
the same rule. And it further follows: 'Socrates exists, there-
fore either Socrates exists or a stick stands in the corner'.
The consequence holds, since from every categorical propo-
sition a disjunctive proposition is deducible (infert) of which
it is a part. And then again: 'Socrates exists and Socrates does
not exist; therefore (by the second part of this copulative
proposition): Socrates does not exist; therefore a stick stands
in the corner'. The consequence holds since the consequence
is formal from a disjunctive with the denial (destrudio) of one
* Omitting necessariam.
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CONSEQUENCES
of its parts to the other. And so every proposition similar in
form to this would be a valid consequence if it were formed.
This rule is usually expressed in the following words : 'from
every copulative consisting of contradictorily opposed parts,
there follows any other (proposition by) formal consequence'.
This text is undoubtedly one of the peaks of scholastic prepo-
sitional logic. Both the rule and its proof were part of the scholastic
capital. It is to be found in Pseudo-Scotus in the form:
31.23 From every proposition evidently implying a con-
tradiction, any other formally follows. So there follows for
instance: 'Socrates runs and Socrates does not run, therefore
you are at Rome.'
The proof in 31.22 relies on the following laws as axioms, which
are expressly formulated :
31.221 If P and Q then P.
31.222 If P and Q then Q.
31.223 If P then, P or Q.
31.224 If P or Q, then, if not-P then Q.
The proof runs thus:
(1) P and not-P (hypothesis)
(2) P (by (1) and 31.221 with sub-
stitution of 'not-P' for lQ')
(3) P or Q (by (2) and 31.223 (cf. 31.10))
(4) not-P (by (1) and 31.222 with sub-
stitution of 'not-P' for lQ')
(5) Q (by (3), (4) and 31.224)
And this was the probandum, Q being any proposition at all.
Of the laws used in this proof, 31.221-2 are to be found in Ockham
(31.24) and were also familiar to Paul of Venice. 31.223 is the modern
law of the factor, accepted by the Scholastics from the time of Kil-
wardby (31.10). 31.224 is the later modus tollendo ponens, ana-
logous to the fifth indemonstrable of the Stoics (22.08), but using
non-exclusive disjunction.
D. PAUL OF VENICE
Next we give some rules for copulative propositions, from Paul
of Venice.
31.25 For the truth of an affirmative copulative (propo-
sition) there is required and suffices the truth of both parts
of the copulative. . . .
31.26 A corollary from this rule is the second: that for the
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SCHOLASTIC LOGIC
falsity of an affirmative copulative the falsity of one of its parts
is sufficient. . . .
31.27 The third rule is this: for the possibility of the copu-
lative it is required and suffices that each of its principal parts
is possible and each is compossible with each - or if there are
more than two, with all. . . .
31.28 From this follows the fourth, viz.: for the impos-
sibility of a copulative it is sufficient and requisite that one of
its principal parts be impossible or that one be not compossible
with the other, or the others. . . .
31.29 The fifth rule is this: for the necessity of an affir-
mative copulative, the necessity of every one of its parts. . . .
31.30 From this rule follows the sixth: that for the contin-
gence of a copulative it is required and suffices that one of its
categorical principal parts be contingent and compossible with
the other, or with all others if there are more than two.
Similar rules for 'known', 'known as true', 'credible' follow.
For disjunctives, the same author gives the following rules,
among others:
31.31 From what has been said (cf. 31.223) there follow
four corollaries. The first is: if there is an affirmative dis-
junctive . . . composed of two categoricals of which one is
superordinate to the other by reason of a term or terms in it,
the argument is valid to the superordinate part; it follows
e.g.: you run or you are in motion, therefore you are in
motion. . . .
31.32 The second corollary is this: if there is a disjunctive
consisting of two categoricals of which one is possible, the
other impossible, the argument to the possible part is valid.
Hence it follows validly: 'there is no God, or you do not exist,
therefore you do not exist'; 'you are an ass or you run,
therefore you run'.
31.33 The third corollary is this: if there be formed a
disjunctive of two categoricals that are equivalent (convertibi-
libus), the argument to each of them is valid, for it follows
validly: 'there is no God or man is an ass, therefore man is an
ass'. And from the same antecedent it follows that there is no
God, since those categoricals, being impossible, are equivalent.
It further follows: 'you are a man or you are risible, therefore
you are risible', and it also follows that you are a man.
31.34 The fourth corollary is this: if a disjunctive be
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CONSEQUENCES
formed of two categoricals of which one is necessary and the
other contingent, the argument is valid to the necessary part.
Hence it follows validly: 'you run or God exists, therefore
God exists'. And it is not strange that all such consequences
hold, for the consequent follows of itself immediately (con-
tinue) from each part of the disjunctive, hence it must follow
too from the disjunctives themselves.
31.35 The eighth principal rule is this: from an affirmative
disjunctive ... to the negative copulative formed of the
contradictories of the parts of the disjunctive is a valid
argument. The proof is that the affirmative copulative formed
of the contradictories of the parts of the disjunctive contradicts
the disjunctive, therefore the contradictory of that copulative,
formed by prefixing a negative, follows from the disjunctive.
For example 'you run or you are in motion, therefore: not,
you do not run and you are not in motion' ; 'God exists or no
man is an ass, therefore: not, there is no God and man is an
ass'. Those consequences are evident, since the opposites
of the consequents are incompatible with their antecedents,
as has been said.
31.36 From that rule there follows as a corollary that from
an affirmative copulative ... to a negative disjunctive
formed of the contradictories of the parts of the copulative is
a valid argument. Hence it follows validly: 'you are a man
and you are an animal, therefore: not, you are not a man or
you are not an animal'. Similarly it follows: 'you are not a
goat and you are not an ass, therefore : not*, you are a goat or
you are an ass'. . . .
The last two texts contain two of the so-called 'de Morgan ' laws.
So far as is known, they first occur in Ockham (31.37) and Burleigh
(31.38)**. The latter gives them in the form of equivalences.
31.39 If one argues from an affirmative conditional,
characterized (denominaia) by 'if, to a disjunctive consisting
of the contradictory of the antecedent and the consequent*
of the conditional the consequence is formal. The proof is
that this consequence is formal : 'if you are a man you are an
animal; therefore, you are not a man or you are an animal'.
And no one example is more cogent than another, therefore all
are valid consequences.
* Adding non.
* * But Peter of Spain ( Traclalus Syncategorematum) has the doctrine of 31.36.
* * * Reading consequente for consequentis.
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SCHOLASTIC LOGIC
E. RULES OF CONSEQUENCES UT NUNC
Buridan writes:
31.40 And it is to be noted that a proportionate conclusion
is to be posited concerning consequences ut nunc (i.e. propor-
tionate to that concerning simple consequences), viz. that
from every false proposition there follows every other by a
consequence ut nunc, because it is impossible that things being
as they now are a proposition which is true should not be true.
And so it is not impossible that it should be true, however
anything else may not be true. And when the talk is of the
past or future, then it can be called a consequence ut nunc, or
however else you like to call it, e.g. it follows by a consequence
ut nunc or ut tunc or even nunc per tunc: 'if Antichrist will not
be generated, Aristotle never existed'. For though it be simply
true that it is possible that Antichrist will not exist, yet it is
impossible that he will not exist when things are going to be
as they will be ; for he will exist, and it is impossible that he
will exist and that he will not exist.
We have here first of all the two classical 'paradoxical' laws of
material implication:
31.401 If P is false then Q follows from P.
31.402 If P is true, then P follows from Q.
Buridan provides an example of (substitution in) the first of
these laws, putting the proposition 'Antichrist will not be generated'
for lP\ and 'Aristotle never existed' for lQ\ The first proposition
is, absolutely speaking, possible, so this cannot be a case of simple
consequence (cf. the first rule, 31.14 supra), for that would require
it to be absolutely impossible. But the consequence holds if taken
ut nunc, since the proposition 'Antichrist will not be generated' will
in fact be impossible in what will be the circumstances. Hence
we have impossibility for that time (ut tunc) and so a consequence
that holds for that time.
This shows that even consequence for a given time is defined by
means of impossibility. The difference between it and simple conse-
quence consists only in the kind of impossibility, absolute (for all
times and circumstances) in the case of simple consequence, con-
ditioned in that of consequence for a given time.
But impossibility ut nunc is defined as simple non-existence, and
so the proposition 'Antichrist will not be generated' can be reckoned
as impossible since Antichrist will in fact be generated. It follows
that consequence ut nunc can be defined without the help of the
modal functor, a proposition ut nunc being impossible simply when
it is false.
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CONSEQUENCES
Another law of consequence ut nunc comes from the same text of
Buridan :
31.41 If a conclusion follows from a proposition together
with one or more additional propositions, the same conclusion
follows from that proposition alone by a consequence ut nunc.
This rule is analogous to the seventh rule given above for simple
consequence (31.21), which shovvs that the whole system of simple
consequences can be transformed into a system of consequences
ut nunc by everywhere replacing 'necessary' by 'true' and 'impossi-
ble' by 'false' and similar simplifications.
Finally we remark that Buridan, so far as is now known, is the
only scholastic logician to develop laws of consequence ut nunc,
though he devoted much less space to them than to simple conse-
quences. In Paul of Venice the subject of consequence ut nunc seems
to have dropped out completely.
209
III. LOGIC OF TERMS
§32. ASSERTORIC SYLLOGISTIC
Contrary to a widespread opinion, the assertoric syllogistic was
not only not the only, it was not even the chief subject of scholastic
logic. The Scholastics, like most of the Commentators (24.271 ff.),
thought of syllogisms as rules (cf. 30.11) rather than conditional
propositions. The domain of syllogisms received a significant exten-
sion through the introduction of singular terms already in Ockham's
time. But the new formulae thus derived will here be given separately
under the heading 'Other Formulae' (§ 34) since they effect an
essential alteration in the Aristotelian syllogistic. In the present
section we shall confine ourselves to that part of the scholastic
treatment which can still be deemed Aristotelian. Here, too,
everything is treated purely metalogically (except in some early
logicians such as Albert the Great), but that is quite in the Aristo-
telian tradition (14.25ff.).
The most important contributions are these: (1) the devising of
numerous mnemonics for the syllogistic moods and their inter-
relationships, culminating in the pons asinorum. (2) The systematic
introduction and thorough investigation of the fourth figure. (3)
The position and investigation of the problem of the null class, which
has already received mention in connection with appellation (§ 28, B).
A. EARLY MNEMONICS
L. Minio-Paluello has recently made the big discovery of an early
attempt to construct syllogistic mnemonics, in a MS of the early
13th century. This deprives of its last claims to credibility Prantl's
legend of the Byzantine origin of such mnemonics. * The essentials
are these:
32.06 It is to be remarked that there are certain notations
(notulae) for signifying the moods. . . . The four letters e, i, o, u
signify universal affirmatives, and the four letters /, m, n, r
signify universal negatives, and the three** a, s, t signify
* Carl Prantl, relying on a single MS, ascribed to Michael Psellus (1018 to
1078/96) the 'Lvvotyit; elq ttjv 'AptaTOTeXou<; Xoyix^v eTuarrjpgv in which such
mnemonics occur, and stated that the Summulae of Peter of Spain was a trans-
lation from that (32.01). In that opinion he was the victim of a great mistake,
since M. Grabmann (32.02) following C. Thurot (32.03), V. Rose (32.04) and
R. Stapper (32.05) has shown the Hvvotyic, to be by George Scholarios (1400-1464)
and a translation of the Summulae.
* * Reading '3' for '4'.
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ASSERTORIC SYLLOGISTIC
particular affirmatives, and b, c, d signify particular negatives.
So the moods of the first figure are shown in the following
verse: uio, non, est (tost), lac, uia, mel, uas, erp, arc. Thus the
first mood of the first figure, signified by the notation uio,
consists of a first universal affirmative and a subsequent
universal affirmative, (and) concludes to a universal affirma-
tive; e.g.: All man is animal and all risible is man, therefore
all risible is animal. . . . The moods of the second figure are
shown in the following verse: ren, erm, vac* , obd. . . . The
moods of the third figure are shown in the following verse:
eua, nee, aut, esa, due, nac.
This is a very primitive technique, but at least it shows that the
highly developed terminology of Peter of Spain*** had antecedents
in Scholasticism itself.
We cite the relevant texts from the Summulae Logicales.
B. BARBARA-CELARENT
32.07 After giving a threefold division of propositions it is
to be known that there is a threefold enquiry to be made about
them, viz. What?, Of what kind?, How much? 'What?'
enquires about the nature (substantia) of the proposition,
so that to the question 'What?' is to be answered 'categorical'
or 'hypothetical'; to 'Of what kind?' - 'affirmative' or
'negative'; to 'How much?' - 'universal', 'particular',
'indefinite', 'singular'. Whence the verse:
Quae ca vel hip, qualis ne vel aff, un quanta par in sin.
the questions being in Latin: quae?, qualis?, quanta?, and
the answers: categorica, hypoihetica, affirmaiiva, universalis,
particularis, infinita, singularis.
So far as we know this is the first text in which the notions of
quality and quantity occur. The full set of technical terms connected
therewith, together with some others, appears in the next passage
which resumes the doctrine of conversion.
32.08 The conversion of propositions having both terms
in common but with the order reversed, is threefold; viz.
* MS has 'rachc\
* * Personal thanks are due to Prof. L. Minio-Paluello for telling us of this MS
and helping to restore the text.
* * * This can not have originated with him. Prof L. Minio-Paluello informed
me on 24. 6. 55 that he had found the word 'Fesiino' in a MS dating at the latest
from 1200.
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SCHOLASTIC LOGIC
simple, accidental, and by contraposition. Simple conversion is
when the predicate is made from the subject and conversely,
the quality and quantity remaining the same. And in this
way are converted the universal negative and the particular
affirmative. . . . The universal affirmative is similarly con-
verted when the terms are equivalent (convertibilibus). . . .
Accidental conversion is when the predicate is made from
the subject and conversely, the quality remaining the same,
but the quantity being changed ; and in this way the universal
negative is converted into the particular negative, and the
universal affirmative into the particular affirmative. . . .
The law of accidental conversion of the universal negative is not
in Aristotle.
32.09 Conversion by contraposition is to make the predi-
cate from the subject and conversely, quality and quantity
remaining the same, but finite terms being changed to
infinite ones. And in this way the universal affirmative is
converted into itself and the particular negative into itself,
e.g. 'all man is animal' - 'all non-animal is non-man'; 'some
man is not stone' - 'some non-stone is not non-man'.*
Hence the verses:
A Affirms, E rEvokes**, both universal,
/ affirms, 0 revOkes**, both in particular.
Simply converts fEel, accidentally EvA,
AstO by contra(position) ; and these are all the conversions.
The classical expressions Barbara, Celarent, etc. seem to have
been fairly generally known about 1250. After describing the
assertoric moods Peter of Spain introduced them thus:
32.10 Hence the verses:
Figure the first to every kind*** concludes,
The second only yields negations,
Particulars only from third figure moods.
Barbara, celarent, darii, ferion, baralipton,
Celantes, dabitis, fapesmo, frisesomorum.
Cesare. camestres, festino, baroco, darapti.
Felapto, disamis, datisi, bocardo, ferison.
* Reading non homo.
* * negat.
* * * viz. problematis.
212
ASSERTORIC SYLLOGISTIC
32.11 In those four verses there are twenty-one expressions
(didiones) which so correspond to the twenty-one moods of the
three figures that by the first expression is to be understood the
first mood, and by the second the second, and so with the others.
Hence the first two verses correspond to the moods of the
first figure, but the third to the moods of the second save for
its last expression. It is to be known therefore that by these
four vowels, viz. A, E, 1,0 set in the aforesaid verses there
are understood the four kinds of proposition. By the vowel A
is understood the universal affirmative, by E the universal
negative, by / the particular affirmative, by 0 the particular
negative.
32.12 Further it is to be known that in each expression
there are three syllables representing three propositions, and
if there is anything extra it is superfluous, excepting M as
will appear later. And by the first syllable is understood the
major proposition, similarly by the second the second proposi-
tion, and by the third the conclusion; e.g. the first expression,
viz. Barbara, has three syllables, in each of which A is set,
and A set three times signifies that the first mood of the first
figure consists of two universal affirmatives concluding to a
universal affirmative; and thus it is to be understood about
the other expressions according to the vowels there set.
32.13 Further it is to be known that the first four expres-
sions of the first verse begin with these consonants, B, C, D,
F, and all the subsequent expressions begin with the same,
and by this is to be understood that all the subsequent moods
beginning with B are reduced to the first mood of the first
figure, with C to the second, with D to the third, with F to
the fourth.
32.14 Further it is to be known that wherever S is put in
these expressions, it signifies that the proposition understood
by means of the vowel immediately preceding should be
converted simply. And by P is signified that the proposition
which is understood by means of the vowel immediately
preceding is to be converted accidentally. And wherever M
is put, it signifies that transposition of the premisses is to be
effected. Transposition is to make the major minor and
conversely. And where C is put, it signifies that the mood
understood by means of that expression is to be reduced
per impossibile.
Whence the verses:
213
SCHOLASTIC LOGIC
S enjoins simple conversion, per accidens P,
Transpose with M, ad impossibile C.
George Scholarios's Greek version of the four last verses of 32.10
is not without interest (cf. 32.02 ff.):
32.15 Tpa^fjiaTa Kypa^e ypacpiSi re^vixoc;, (I)
rpa[X(JLacr!,v fe'ra^e x^?lGl rcapOevos ispov (la)
"Eypoc^s y.oi'zzjz uiTpiov a^oXov. (II)
"Ktzolgi aOevapcx; taaxt? a<j7ci§i 6(xaXo<; (pepwjTos. (Ill)
Unlike the Latin ones, these verses are meaningful, and can be
rendered:
Letters there wrote with a style a scholar,
With letters there composed for the Graces a maiden a dedica-
tion.
She wrote: Cleave to the moderate, un-wrathful (man).
In all, that strength which like a shield is well-proportioned is
the best.
The names Barbara, Celareni etc. have survived the era of Scho-
lasticism and are still in use today, unlike many other syllogistic
mnemonics. We give some examples of these others, and first some
which concern the technique of reducing moods of the second and
third figures (also of the 'indirect' moods of the first figure : 17.111 ff.)
to moods of the first. For that purpose, Jodoc Trutfeder at the
beginning of the 16th century gave these expressions:
32.16 Baralipton Nes- Celareni
Celantes
ci-
Darii
Dab His
e-
Celareni
Fapesmo
ba-
Barbara
Frisemom
tis.
Darii
Cesare
0-
Ferio
Camestres
di-
Darii
Festino
e-
Celareni
Baroco
bam.
Barbara
Darapli
Le-
Celareni
Felapton
va-
Barbara
Disamis
re.
Celareni
Datisi
Bo-
Ferio
Bocardo
man-
Barbara
Ferison
nis.
Darii
These expressions serve for the indirect process of Aristotle
(§ 14, D). Thus for instance from Celareni, by putting the contra-
dictory opposite (/) of its conclusion (E) for the minor premiss (A)
there is concluded the contradictory of the latter (0), and one has
214
ASSERTORIC SYLLOGISTIC
Festino. Applying this treatment to the major premiss one has
Disamis. Evidently some further processes must be employed to
get a few further moods. E.g. to get Felapton and Darapti from
Barbara and Celarent respectively, one must first deduce Barbari
and Celaronl (24.271 f.). So too in the case of Dabilis (24.273).
Further mnemonics that were used will he mentioned later
(32.24 and 32.38).
C. BARBARI-CELARONT
Similar mnemonic expressions are found for the so-called 'sub-
alternate' moods (24.271-24.281). A complete list with names
appears in a text of Peter of Mantua that is in other respects very
defective:
32.17 . . . the first (formula) ... is usually signified by the
expression Barbara. The second formula has premisses arranged
in the way described which conclude to the particular affir-
mative or indefinite of the consequent of the first formula
that we posited, and this we are wont to call Barbari. . . .
From the aforesaid (premisses) can also be concluded the
particular negative of (i.e. corresponding to) the consequent
of the aforesaid formula, which (new) formula we can call
Celaronl. . . .
The eighth formula, which is called Baralipton, follows
from Barbari, by conversion of its conclusion. . . .
The ninth formula is called Celantes . . . from it follows the
tenth, which is called Celantos; it concludes to a particular
or indefinite conclusion. . . .
The second mood (of the second figure) can be gained from
the aforesaid premisses (of the mood Cesare) by concluding
to the particular that corresponds to (Cesare's) conclusion,
and is signified by the expression Cesaro. . . .
The next formula ... is usually called Cameslres. From it
there follows another formula which we call Camestro.
So Peter of Mantua has five subalternate moods besides the
nineteen moods of Peter of Spain, in fact the full twenty-four. But
he has many others as well, commonly forming an 'indirect' mood
corresponding to each of the others (applying the Aristotelian rules
of § 13), e.g. a Cesares corresponding to Cesare. Cesares would look
like this:
No man is stone ;
All marble is stone ;
Therefore no man is marble.
215
SCHOLASTIC LOGIC
Contrast the following in Camestres :
All marble is stone;
No man is stone;
Therefore no man is marble.
The only difference between the two is in the order of the premisses,
and to reckon them as distinct moods is an extreme of formalism.
Peter of Mantua further forms such moods as Barocos, with the
O-conclusion of Baroco converted (!) and other false formulas.*
D. THE FOURTH FIGURE
1. Among the Latins
We know of no scholastic logical text in Latin where the fourth
figure in the modern sense can be found, though all logicians of the
period develop the 'indirect moods of the first'. They are mostly
aware of a fourth figure, but treat it as not distinct from the first, e.g.
Albert the Great (32.18), Shyreswood (32.19), Ockham (32.20),
Pseudo-Scotus (32.21), Albert of Saxony (32.22), Paul of Venice
(32.23). We give an instance from Albert of Saxony.
32.22 (The syllogism is constituted) in a fourth way if
the middle is predicated in the first premiss, subjected in the
second. . . . But it is to be noted that the first figure differs
from the fourth only by interchange of premisses which does
nothing towards the deducibility or non-deducibility of the
conclusion.
Some later logicians do recognize a 'fourth figure' but this again
is not the modern one; only the first with interchanged premisses, as
in the last text. This is very clear in Peter Tarteret and Peter of
Mantua. We quote the first:
32.24 First (dictum): Taking 'figure' in a wide sense, the
fourth figure is no different from the first but contained under
it. Second: taking 'first figure' in a specific sense, a fourth
figure is to be posited distinct from the first; and the fourth
figure consists in this, that the middle is predicate in the
major premiss, subject in the minor, e.g. 'all man is animal;
all animal is substance; therefore all man is substance'.
* It should be understood in respect of this and the following sub-section
that after Peter of Spain (generalizing the method of Boethius and Shyreswood
for the second and third figures) it was usual to define the major and minor
premisses as the first and second stated, and the extreme terms with reference
to the premisses, not the conclusion. 48 moods in 4 figures can be (and some-
times were) correctly distinguished on this basis. 'Classical' failure to distinguish
this from the method of Albalag (which goes back to Philoponus) resulted in
many inconsistencies. (Ed.)
216
ASSERTORIG SYLLOGISTIC
Third : there are four moods of the fourth figure, viz. Bamana,
Camene, Dimari, and Fimeno. They are reduced to the first
figure by mere exchange of premisses.
Peter of Mantua also has Bamana etc.
2. In Albalag
Yet a clearly formulated doctrine of the 'genuine' fourth figure is
to be found in a 13th century text- of the Jewish philosopher Alba-
lag.* This text, like the foregoing, seems to have been without
influence on the development of logic in the Middle Ages. It was
recently discovered by Dr. G. Vajda and has never been translated
into Latin. We quote it here at length for its originality, and because
it is instructive about the level of logic at that time.
32.25 In my opinion there must be four figures. For the
middle term can be subject in one of the two premisses and
predicate in the other in two ways: (1) the middle term is
subject in the minor, predicate in the major, (2) it is predicate
in the minor, subject in the major. The ancients only con-
sidered the second arrangement and called it the 'first figure'.
This admits of four moods which can yield a conclusion. But
the first arrangement, which I have found, admits of five
moods which can yield a conclusion. . . .
32.26 We say then that this new figure is subject to three
conditions: (1) one of its premisses must be affirmative, the
other universal; (2) if the minor premiss is affirmative, the
major will be universal; (3) if the major is particular, the
minor will be affirmative.
The conditions exclude eleven of the sixteen (theoretically
possible) moods; there remain therefore five which can yield
a conclusion.
32.27 (1) The minor particular affirmative, the major
universal negative :
Some white is animal.
No raven is white.
Some animal is not raven.
Then one can convert the minor particular affirmative and
the major universal negative and say:
Some animal is white,
No white is raven,
which yields the third mood of the first figure.
* This was kindly put at my disposal, together with a French translation, by
Dr. G. Vajda.
217
SCHOLASTIC LOGIC
32.28 (2) The minor universal affirmative, the major
universal negative:
All man is animal.
No man is raven.
Some animal is not raven.
Then one can get back to the third mood of the first figure
by converting both premisses.
32.29 (3) The minor universal negative, the major universal
affirmative :
No man is stone.
Every speaker is man.
No stone is speaker.
Exchanging the minor and major with one another, one comes
back to the second mood of the first figure, of which the
conclusion will be: 'No speaker is stone', and one only needs
to convert this to obtain 'No stone is speaker'.
32.30 (4) Two affirmatives:
All composite is not eternal.
All body is composite.
Some not eternal is body.
Here one can interchange the minor and major premisses
and reach the first figure, with conclusion: 'All body is not
eternal' which can be converted to 'Some not eternal is body'.
32.31 (5) The minor universal affirmative, the major
particular affirmative :
All man is speaker.
Some white is man.
Some speaker is white.
If one interchanges the minor and major premisses, one
concludes in the first figure: 'some white is speaker' which
will be converted as above. . . .
32.32 . . . the syllogism is formed with reference to a
determinate proposition which is first established and laid
down in the mind, and the truth of which one then tries to
justify and manifest by means of the syllogism. Of the
premisses, that containing the term which is predicate of
this proposition is the major, that containing the subject
is the minor.
Albalag here presents the modern definition of the syllogistic terms,
not according to their extension, but formally, according to their pla-
ces in the conclusion. The modern names of the moods he introduces
are: Fresison, Fesapo, Calemes, Bamalip and Dimaris (cf. § 36, F).
218
assertory syi.i.oojsth;
There is missing only that corresponding to Peter of Mantua's
Celantos (32.17), viz. Calemop. Albalag also formulates the general
rules of the fourth figure, and uses the combinatorial method.
E. COMBINATORIAL METHOD
In Albert the Great we find a procedure taken over from the
Arabs (32.33) by which all possible moods of the syllogism are first
determined combinatorially, and the invalid ones then discarded.
The relevant text runs:
32.34 It is to be known that with such an ordering of
terms and arrangement of premisses (propositionum) sixteen
conjugations result, yielded by the quantity and quality
of the premisses. For if the middle is subject in the major and
predicate in the minor, either (1) both premisses are universal,
or (2) both are particular, or one (is) universal and the other
particular and this in two ways: for either (3) the major is
universal and the minor particular, or (4) conversely the
major particular and the minor universal; these are the four
gained by combinations of quantity. When each is multiplied
by four in respect of affirmation and denial, there are sixteen
conjugations in all, thus: if both (premisses) are universal
either (1) both are affirmative, or (2) both negative, or (3)
the major is affirmative and the minor negative, or con-
versely (4) the major is negative and the minor affirmative:
and there are four conjugations. But if both are particular,
there are again four conjugations . . . etc.
We may compare with that the text of Albalag (32.25 ff.). Kil-
wardby uses similar methods.
F. INVENTIO MEDII, PONS ASINORUM
The Aristotelian doctrine of the inveniio medii (14.29) was keenly
studied by the Scholastics, and the schema of Philoponus (24.35) was
not only taken over, but also further developed. It is to be found
as early as Albert the Great, who probably found it in Averroes
(32.35) ; Albert's version differs from that of Philoponus and Aver-
roes only in the particular formulae employed. But as we find it in
him, it became the foundation of the famous pons asinorum. so
that it must be given in this form as well :
219
SCHOLASTIC LOGIC
32.36
All Pis M.) AB
(NoPisM.) AD
(All Mis P.) AC
FE (All S is M.
HE (NoSisM.)
(All MisS.)
The further development of this figure is the pons asinorum,
which must have been known to George of Brussels (32.37) since
Thomas Bricot in a commentary on George's lectures gives the
mnemonic words for it with the following explanation:
32.38 When the letters A, E, I, 0 are put in the third
syllable they signify the quality and quantity of the con-
clusion to be drawn. . . . When the letters A and E are put
in the first or second syllable, A signifies the predicate and E
the subject. And each of the letters can be accompanied by
three consonants; A with B, C, D, and then B signifies that
the middle should follow on the predicate, C that it should
be antecedent, D that it should be extraneous. Similarly E
is accompanied by F, G, H, and then F signifies that the
middle should follow on the subject, G that it should be
antecedent, H that it should be extraneous. As is made
clear in these verses:
jB's the subject, F its sequent, G precedent, F outside;
^4's the predicate, B its sequent, C precedent, F outside.
. . . To conclude to a universal affirmative, a middle is to
be taken which is sequent to the subject and antecedent to
the predicate ; and this is shown by Fecana. ... To conclude
a particular affirmative in Darapti, Disamis and Dalisi, a
middle is to be taken which is antecedent to both extremes,
as is made clear by Cageti. ... To conclude to a universal
negative in Celarent or Cesare a middle is to be taken which
220
frunpergUiitaunipti y ■ iituuti
Pontrm u-.*Jrjura (v'rrtrrtuiido cidci't
Impcdichicoo.l •* fenfm firm tc Jjc oc a!tog
In doctis ulcus cl't ilhi nulla lalus
Horrrf rqaus talcm ui ;J.n i nc »itUl?«aC1
Dun gradirurccrnrri' ftc 'kc ire potent
Nod iguur rurfum dira ueuant ^finorum
(^.ui (cdcos retro nunc rcminrrcuolo.
■fhedtutum
/Mmus . mratjboin.dump^ns.opanerrrmu Aims, Hcti mrqd feciSruo nrcfbftmihiqie^
Kficns 5»tiiUiKuadAlab«,uratoicgtcdia* cadca Aux.l.uin.mon.corqua.dcdttWFOiOf.
Pons Asinorum after Peter Tarteret (32.39
ASSERTORIC SYLLOGISTIC
is extraneous to the predicate and consequent on the subject,
as is made clear by Dafenes. But if the inference is to be in
Cameslres, the middle must be extraneous to the subject and
consequent on the predicate, as is made clear by Hebare. . . .
For concluding to a particular negative in the third figure,
the middle should be antecedent to the subject and extra-
neous to the predicate, as is made clear by Gedaco. ... To
conclude indirectly to a particular affirmative, the middle
must be antecedent to the subject and consequent on the
predicate, as is made clear by Gebali.
The mnemonic expressions given here employ the letters of
Albert's figure, so that each expression corresponds to a line of the
figure having the syllables of the mnemonic at its ends, a further
syllable being added to show the quantity and quality of the
corresponding conclusion. So this text of George's both illustrates
the pons asinorum and helps to explain Albert's figure.
The pons asinorum itself we have found only in Peter Tarteret,
with the following introduction:
32.39 That the art of finding the middle may be easy,
clear and evident to all, the following figure is composed
(ponitur) to explain it. Because of its apparent difficulty it
is commonly called the 'asses' bridge' (pons asinorum) though
it can become familiar and clear to all if what is said in this
section (passu) is understood.
We give the schema itself in the preceding illustration.
G. THE PROBLEM OF THE NULL GLASS
The problem of the null class, i.e. of the laws of subalternation,
accidental conversion (14.12, 32.08) and the syllogistic moods
dependent on them, has been much discussed in recent times
(46.01 ff.). It was already posed in the fourteenth century and solved
by means of the doctrines of supposition and appellation. We give
three series of texts, the first attributed to St. Vincent Ferrer, the
second from Paul of Venice, the third from a neo-scholastic of the
17th century, a contemporary of Descartes, John of St. Thomas.
Each gives a different solution.
1. St. Vincent Ferrer
32.40 Under every subject having natural supposition
copulative descent can be made, with respect to the predicate,
to all its supposita, whether such a subject supposes discretely*
* Reading discrete for difinite.
221
SCHOLASTIC LOGIC
or particularly or universally. Therefore it follows validly:
Man is risible, therefore this man is risible, and that
man. . . .
32.41 But against this rule there are many objections. . . .
(Sixth objection:). ... In the propositions 'rain is water
falling in drops', 'thunder is a noise in the clouds', the subjects
have natural supposition. Yet it is not always permissible to
descend in respect to the predicate to the supposita of the
subject; for it does not follow: 'rain is water falling in drops,
therefore this rain, and that rain, etc.'; since the antecedent
is true even when there is no rain, as will be shortly said, and
yet the consequent is not true nor even very intelligible,
since when there is no rain (nulla pluvia exislenle) one cannot
say 'this rain' or 'that rain', since a contradiction would be
at once implied. In the same way must be judged the proposi-
tion 'thunder is a noise etc'
32.42 To the sixth objection it is to be said that that rule . . .
is understood (to hold) when such a subject has supposita
actually (in aclu) and not otherwise. For no descent can be
made to the supposita * of anything except when it has them
actually, since, as the objection rightly says, an evident con-
tradiction would be implied. . . . Hence the consequence which
concludes from a universal proposition to singular ones
contained under it, e.g. 'every man runs, therefore Socrates
runs and Plato runs', and 'every man is an animal, therefore
Plato is an animal etc.', is called by some logicians 'conse-
quence ul nunc1. And rightly so, since no such consequence is
valid except for a determinate time, i.e. when Socrates and
Plato and the other supposita actually exist.
32.43 Against the seventh objection it is to be briefly
said that the subject of the proposition 'the rose is sweet-
smelling' - or as one can also put it 'the rose smells sweet' - has
personal supposition, and it follows validly 'therefore the rose
exists (est)'. But if one says 'the rose is odoriferous' so that
'odoriferous' (hoc quod dicitur odorifera) expresses aptitude,
then the subject has natural supposition and it does not
follow: 'therefore the rose exists'. Hence being odoriferous is
to the rose as living to mankind, and what has been said about
the proposition 'man is living' must also be understood about
this: 'the rose is odoriferous'.
* Reading supposita for subjecla.
222
ASSERTORIC SYLLOGISTIC
The solution of the first text (32.40-42) consists then precisely in
the exclusion of the null class (cf. § 40, Bj, 'null' being taken as
'actually null'. In other words: in the syllogistic every term must
have appellation in the sense of Peter of Spain (28.13 f.). In the
second part (32.43) it is stipulated for subalternation that the subject
must have personal (27.17 f.) and not natural (27.14) supposition.
This evidently presupposes that a term with personal supposition
stands for really existent things. Thus we have the same solution
as before,
2. Paul of Venice
32.44 The third rule is this: universal affirmative and
particular or indefinite (affirmative, as also universal negative
and particular or indefinite) negative (propositions) which
have similarly and correctly supposing terms, are subalternate,
and conversely, explicitly or implicitly, in the logical square
(figura). Hence the following are subalternate: 'every man is
an animal' and 'a certain man is an animal', and similarly:
'no man is an animal' and 'a certain man is not an animal'.
I say 'correctly supposing terms', since the extremes must
explicitly or implicitly stand for just the same thing, if it is a
case of only one suppositum, for the same things, if it is a
case of several. And so I say that (the following) are not
subalternate: 'every man is an animal', 'a certain man is an
animal', since under the supposition that there are no men,
the universal would be true, but the particular false, contrary
to the nature of subalterns. The reason why these are not
subalternate is that the subjects do not stand for exactly
the same thing. The subaltern of the former is therefore : 'man
is an animal', and if one required a particular it must be this:
'a certain being which is a man is an animal'.
So Paul of Venice confines himself to stating the general rule
that both propositions in a subalternation must have subjects with
exactly the same supposition.
3. John of St. Thomas
John of St. Thomas deals with the problem of conversion.
32.45 Against the conversion of the universal affirmative,
it is objected: the consequence 'every white man is a man,
therefore a certain man is a white man' does not hold. For
the antecedent is necessary, but the consequent can be false,
in case no man in the world was white. . . .
223
SCHOLASTIC LOGIC
The answer is that this (proposition) is not true in the sense
in which the first proposition of which it is the converse is
true. For when it is said: 'every white man is a man', with
'is' taken accidentally, for an existing man, this proposition
in the argument given as an example is false, and its converse
too. But when the 'is' abstracts from time and renders the
proposition necessary, then 'white' will not be verified in
the sense of existence, but according to possibility, i.e.
independently of time, in the following sense: 'every possibly
white man is a man', presupposing that no such exists.
Accordingly the converse must be: 'therefore a certain man
is a man who is possibly white', and thus this is true.
The following may serve as explanation: take the proposition (1)
'every Swiss king is a man'. By the rules of conversion (14.11,
32.08) we may infer (2) 'a certain man is a Swiss king'. But (1) is
true, (2) false. Therefore the rule of conversion employed is not
valid. To this the Scholastics would answer that in (1) 'man' evi-
dently stands for a possible man, not for a real one; it has therefore
no appellation in the sense of 28.13. And so if (1) is converted into
(2), 'man' in (2) also supposes for possible men and in this sense (2)
is true as well as (1).
A further interesting point is that singular terms always have
appellation (28.13), so that the Scholastics attribute to proper
names the same property with which the moderns endow descrip-
tions (cf. § 46).
§33. MODAL SYLLOGISTIC
The history of scholastic modal syllogistic has been investigated
from the modern point of view up to and inclusive of Pseudo-Scotus
(33.01). We know that there was more than one system of modal
logic in the Middle Ages and can to some extent follow the devel-
opment.
A. ALBERT THE GREAT
The work of Albert the Great constitutes the starting point and,
as his own text suggests (33.02), would seem to have drawn on
Arabic sources. To begin with, he shows much the same doctrine
as has been ascribed above (29.09) ; cf. also 23.10, 29.12) to Thomas
Aquinas and which is basic for the whole of Scholasticism (33.03),
viz. the distinction of the composite (composita, de dido) and the
divided (divisa, de re) modal proposition, i. e. between that in which
the modal functor governs the whole dictum and that in which it
224
MODAL SYLLOGISTIC
governs only a part. Later Albert gives a clear statement of the
Aristotelian distinction of the two structure- of the modal proposi-
tion in the divided sense:
33.04 That the predicate A possibly belongs to the subject
B means one of these: (1) that A possibly belongs to that
which is B and of which B is predicated in the sense of actual
inherence, or (2) that A possibly belongs to that to which B
possibly belongs.
There is added a third structure, unknown to Aristotle:
33.05 And if someone asks why the third meaning (acceptio)
of the contingent is not given here, viz. that whatever is
necessarily B is possibly A, since this is used in the mixing of
the contingent and the necessary, it is to be answered that it
is left sufficiently clear from what else has been determined
about the mixing of the assertoric and the contingent.
The structure in question is this:
For all x: if x is necessarily B, x is possibly A.
It is significant that Albert the Great puts this doctrine at the
beginning of the presentation of his theory of modal syllogisms in a
special chapter entitled De dici de omni el dici de nullo in propo-
silionibus de contingenti (33.06). What for Aristotle are marginal
thoughts about the structure of modals by comparison with his
main ideas (15.13), have here become fundamental.
We find then in Albert the Great a systematization of the Ari-
stotelian teaching about the kinds of modal functors (33.07), and a
thorough presentation of the syllogistic of the Prior Analytics.
B. PSEUDO-SCOTUS
Besides the four classical modal functors Pseudo-Scotus intro-
duces others: 'of itself (per se), 'true', 'false', 'doubtful' (dubium),
'known' {scitum), 'opined' (opinatum), 'apparent', 'known' (notum),
'willed' (volitum), 'preferred' (dilectum) (33.08), and so a number of
'subjective' functors. He formulates a long series of laws of modal
propositional logic (modal consequences), among which are the
following:
33.09 If the antecedent is necessary the consequent is
necessary . . . and similarly with the other (positive) modes.
33.10 Modal (de modo) propositions in the composite sense
with the (negative) modes 'impossible', 'false', 'doubtful' are
not convertible like assertoric ones. Proof: for otherwise the
225
SCHOLASTIC LOGIC
(following) rules would be true : 'if the antecedent is impossible,
the consequent is impossible', 'if the antecedent is doubtful,
the consequent is doubtful' ; but they are false. . . .
33.11 It follows: possibly no B is A, therefore possibly no
A is B: since both (propositions) 'no B is A' and 'no A is B'
follow from one another. So if one is contingent, the other is
too: otherwise the contingent would follow from the neces-
sary. . . .
33.12 If the premisses are necessary, the conclusion is
necessary.
With the help of these and other laws known to us from the
chapter on propositional consequences, Pseudo-Scotus proceeds
to establish two systems of syllogistic, one with modal propositions
in the composite, the other in the divided sense. As premisses he
uses not only contingent but also (one-sidedly: (15.071) possible
and impossible propositions. We cite only a few examples from his
teaching on conversion:
33.13 Modal proposition in the composite sense are con-
verted in just the same way as assertoric ones.
33.14 Affirmative possible (de possibili) propositions in the
divided sense (in which the subject stands) for that which is,
are not properly speaking converted. Proof: on the supposition
that whatever is in fact running is an ass, the following is
true : 'every man can run' in the sense that everything which
is a man is able to run, but its converse is false: 'a certain
runner can be a man', . . . And I say 'properly speaking' on
purpose, since (these propositions) can in a secondary sense
be converted into assertorics. E.g. 'every man can run,
therefore a certain thing that can run is a man'. . . .
33.15 The third thesis concerns affirmative possible pro-
positions, in which the subject stands for that which can
exist, for such affirmatives are converted in the same way as
assertorics. . . .
33.16 As concerns necessary propositions, and first those
which are to be understood in this (divided) sense with subject
standing for what is, the first thesis is this, that affirmatives. . .
are not converted; for supposing that God is creating, it
does not follow: 'whatever is creating is necessarily God,
therefore a certain God is creating necessarily'.
33.17 The second thesis is that negative necessary propo-
sitions with subject standing for what is, are not converted. . . .
226
MODAL SYLLOGISTIC
33.18 Third thesis: that affirmative necessary propositions
with subject standing for what can be are not properly
speaking converted. . . .
33.19 Fourth thesis about necessary propositions with
subject standing for what can be : universal negatives are
simply converted, particular negatives not. Proof: since, as
has been said earlier, the particular affirmative possible
(proposition with subject standing) for what can be, converts
simply, and it contradicts the universal negative necessary
(proposition with subject standing) for what can be; so: if
one of two contradictories is simply converted, so is the other,
since when the consequent follows from the antecedent, the
opposite of the antecedent follows from the opposite of the
consequent.
c. OCKHAM
Pseudo-Scotus introduced 1. one-sidedly possible premisses into
the syllogistic, 2. in the composite sense. Ockham has a further
innovation: he treats also of syllogisms in which one premiss is
taken in the composite, the other in the divided sense. At the same
time the whole modal syllogistic is formally developed from its
structural bases with remarkable acumen. We give only two
examples:
33.20 As to the first figure it is to be known that when
necessary premisses are taken in the composite sense, or
when some are taken that are equivalent to those propositions
in the composite sense, there is always a valid syllogism
with a conclusion that is similar in respect of the composite or
equivalent sense. . . . But when all the propositions are taken
in the divided sense, or equivalent ones, a direct conclusion
always follows, but not always an indirect. The first is evident
because every such syllogism is regulated by did de omni or
[did) de nullo. For by such a universal proposition it is denoted
that of whatever the subject is said, of that the predicate is
said. As by this: 'every man is necessarily an animal' is
denoted that of whatever the subject 'man' is said, of that the
predicate 'animal' is necessarily said. And the same holds
good proportionately of the universal negative. Therefore
adjoining a minor affirmative in which the subject (of the
major) is predicated of something with the mode of necessity,
the inference proceeds by did de omni or de nullo (cf. 14.23).
Hence it follows validly: 'every man is necessarily an animal,
227
SCHOLASTIC LOGIC
Socrates is necessarily a man, therefore Socrates is necessarily
an animal'. But the indirect conclusion, viz. the converse of
that conclusion with no other variation than the transposition
of terms does not follow . . . (There follows here the reason as
in 33.16). But if the major is taken in the composite sense,
or an equivalent (proposition), and the minor in the divided
sense, the conclusion follows in the composite sense and not in
the divided. The first is evident, because it follows validly:
'This is necessary: every divine person is God, one creating is
necessarily a divine person, therefore one creating is necessa-
rily God'. But this does not follow: 'therefore this is necessary:
one creating is God'.
But if the major is taken in the divided sense, or an equi-
valent (proposition), and the minor in the composite sense,
the conclusion follows in the divided sense and in the composite
sense. And the reason is that it is impossible that something
(B) essentially (per se) or accidentally inferior (to A) should
be necessarily predicated of something (67), so that the
proposition ('67 is B') would be necessary, without the
proposition in which the superior (A) of that inferior (B) is
predicated of the same (67) (viz. '67 is A') also being neces-
sary.
So for every Aristotelian formula Ockham has the four: (1) with
both premisses in the composite sense, like Theophrastus (cf.
17.15 f.); (2) with both premisses in the divided sense, like
Aristotle - as was concluded from the indications available (§ 15, B) ;
(3) the major premiss composite, the minor divided ; (4) the major
premiss divided, the minor composite.
Another example is the following treatment of syllogisms with
both premisses in the mode of simple (one-sided) possibility:
33.21 ... I here understand 'possible' in the sense of the
possibility which is common to all propositions that are not
impossible. And it is to be known that in every figure, if all
the propositions be taken as possible in the compounded
sense, or if equivalent ones to those be taken, the syllogism is
invalid because inference would proceed by this rule: the
premisses are possible, therefore the conclusion is possible,
which rule is false. Hence it does not follow: 'that everything
coloured is white is possible, that everything black is coloured
is possible, therefore that everything black is white is pos-
sible'. . . . And so the rule is false: the premisses are possible
228
MODAL SYLLOGISTIC
therefore the conclusion is possible. But this rule is true: if
the premisses are possible and compossible, the conclusion is
possible (cf. 31.27) . . . But if the possible proposition be taken
in the divided sense, or an equivalent one be taken, such as are
propositions like 'every man can be white', 'a white thing can
be black' etc. . . . there the subject can stand for things
which are or for things which can be, i.e. for things of which it
is verified by a verb in the present, or for things of which it is
verified by a verb of possibility. ... As if I say: 'every white
thing can be a man', one sense is this: everything which is
white can be a man, and this sense is true if there be nothing
white but man. Another sense is this: everything which can
be white can be a man, and this is false whether only man be
white or something other than man. . . .
And it is to be known that if the subject of the major be
taken for things which can be . . . however the subject of the
minor be taken, the uniform syllogism is always valid and is
regulated by did de omni or de nullo, and the common prin-
ciples of the assertoric syllogism hold. E.g. if one argues thus:
'every white thing can be a man - i.e. everything that can be
white can be a man - every ass can be white, therefore every
ass can be a man'. . . .
But if the subject of the major supposes for things which
are, then such a uniform (syllogism) is not valid, for it does not
follow: 'everything which is white can be a man, every ass
can be white, therefore every ass can be a man'. For if there
be nothing white but man, the premisses are true and the
conclusion false. . . .
These examples may suffice to give an idea of the problems
discussed.
We now give a summary of the different kinds of modal syllogism
which Ockham considered. He distinguishes the following functors
and kinds of functor: (1) 'necessary', (2) 'possible' (one-sidedly),
(3) 'contingent' (two-sidedly, (4) 'impossible', (5) other modes
(subjective). Further there are (6) the assertoric propositions. Ock-
ham deals with syllogisms with premisses in the following combi-
nations:
3-5
229
1-1
6-1
2-2
6-2
1-2
3-3
6-3
1-3
2-3
4-4
6-4
1-4
2-4
5-5
6-5
1-5
2-5
SCHOLASTIC LOGIC
Altogether then he has eighteen classes. In each he discusses the
four formulae mentioned above, and this in each of three figures - the
analogates therefore of the nineteen classical moods. Theoretically
this gives 1368 formulae, but many of them are invalid.
Here, however, as with Aristotle (§ 15, D), there are also many
moods without analogues in the assertoric syllogistic, so that the
total number of valid modal syllogisms for Ockham, in spite of the
many invalid analogues, may reach about a thousand.
D. LOGIC OF PROPOSITIONS IN FUTURE AND PAST TENSES
The Scholastics did not look on propositions about the future
and the past as modal, but they treated them quite analogously to
modals. Two texts from Ockham illustrate this point:
33.22 Concerning the conversion of propositions about the
past and the future, the first thing to be known is that every
proposition about the past and the future, in which a common
term is subject, is to be distinguished ... in that the subject
can suppose for what is or for what has been, if it is a proposi-
tion about the past . . . e.g. 'the white thing was Socrates' is
to be distinguished, since 'white' can suppose for what is
white or for what was white. But if the proposition is about
the future, it is to be distinguished because the subject can
suppose for what is or for what will be. . . . Secondly it is to be
known that when the subject of such a proposition supposes for
what is, then the proposition should be converted into a
proposition about the present, the subject being taken with
the verb 'was' and the pronoun 'which', and not into a
proposition about the past. Hence this consequence is not
valid : 'no white thing was a man, therefore no man was white',
if the subject of the antecedent be taken for what is. For let
it be supposed that many men both living and dead have been
white, and that many other things are and have been white,
and that no man is now white, then the antecedent is true and
the consequent false. . . . And so it should not be converted as
aforesaid but thus: 'no white thing was a man, therefore
nothing which was a man was white'.
Then an example from syllogistic :
33.23 Now we must see how syllogisms are to be made
from propositions about the past and the future. Here it is
to be known that when the middle term is a common term, if
the subject of the major supposes for things which are, the
230
OTHER FORMULAE
minor should be about the present and not the future or the
past; for if the minor proposition was about the past and not
the present such a syllogism would not be governed by did de
omni or de nullo, because in a universal major about the past
with subject supposing for things which are, it is not denoted that
the predicate is affirmed or denied by the verb in the past about
whatever the subject is affirmed of by the verb in the past. But
it is denoted that the predicate is affirmed or denied by the verb
in the past about whatever the subject is affirmed of by a verb
in the present. . . . But if the subject of the major supposes for
things which have been, then one should not adjoin a minor
about the present, because as is quite evident, the inference
does not proceed by did de omni or de nullo; but a minor about
the past should be taken, and it makes no difference whether
the subject of the minor supposes for things which are or
things which have been. Hence this syllogism is invalid:
'every white thing was a man, an ass is white, therefore an
ass was a man'. . . . What has been said about propositions
concerning the past, is to be maintained proportionately for
those about the future.
These principles are then applied to the syllogisms in the different
figures.
§34. OTHER FORMULAE
In view of what we know about e.g. the composite and divided
senses (29.13), and of our occasional discoveries of similar doctrines
(28.15ff.), we must suppose that the Scholastics developed a num-
ber of logical theories not pertaining either to propositional logic or to
syllogistic in the Aristotelian sense. But this field is hardly at all
explored ; e.g. we do not know whether they were acquainted with a
more comprehensive logic of relations than that of Aristotle.
We cite a few texts belonging to such theories, viz. (1) a series of
texts about non-Aristotelian 'syllogisms' with singular terms, (2) an
analysis of the quantifiers 'ever' and 'some', (3) a 'logical square'
of so-called 'exponible' propositions, i.e. of propositions equivalent
to the product or sum of a number of categoricals. In that connec-
tion we finally give some theorems about the so-called syllogismus
obliquus, which was not without importance for the later history of
logic.
Here it must be stressed even more than usual, that these are
only fragments concerning a wide range of problems that has not
been investigated.
231
SCHOLASTIC LOGIC
A. SYLLOGISMS WITH SINGULAR TERMS
A first widening of the Aristotelian syllogistic consists in the
admission of singular terms and premisses.* Ockham already knows
of the substitution that was to become classic :
34.01 Every man is an animal;
Socrates is a man;
Therefore, Socrates is an animal.
Here the minor premiss is singular. But Ockham also allows
singular propositions as major premisses:
34.02 For it follows validly (bene): 'Socrates is white,
every man is Socrates, therefore every man is white'. . . . And
such a syllogism ... is valid, like that which is regulated by
did de omni or de nullo, since just as the subject of a universal
proposition actually stands for all its significates, so too the
singular subject stands for all its significates, since it only
has one.
The difference between a syllogism as instanced in 34.01 and the
classical Aristotelian syllogism is only 'purely verbal' (34.03)!
This may well be termed a revolutionary innovation. Not only are
singular terms admitted, contrary to the practice of Aristotle, but
they are formally equated with universal ones. The ground advanced
for this remarkable position is that singular terms are names of
classes, just like universal terms, only in this case* unit-classes. There-
fore 34.02 is not propounding the syllogism as a substitution in the
rule:
34.021 If 'for all x: if x is an S then x is a P' holds, and
la is an S' holds, then it also holds : 'a is a P'.
- where 'S' and 'P' are to be thought of as class-names, 'a' as an
individual name -, but as a substitution in:
34.022 If 'for alia;: if x is an M then x is a P' holds, and 'for
all x: if x is an 5 then x is an M' holds, then it holds: 'for all
x: if x is an S then x is a P'
- where 'M', 'S' and 'P' are all class-names. In that case the sole
difference between the Aristotelian and Ockhamist syllogisms is that
the former is a proposition, the latter a rule. Admittedly the basis of
the system is altered with the introduction of names for unit-
classes.
Again, the syllogisms with singular terms that are usually attri-
buted to Peter Ramus, are already to be found in Ockham.
34.02 contains one example; here is another:
* But 34.01 is Stoic. Vid. Sextus Empiricus, Pyrr. Hyp. B 164 ff.
232
OTHER FORMULAE
34.04 Although it has been said above that one cannot argue
from affirmatives in the second figure, yet two cases are to be
excepted from that general rule. The first is, if the middle term
is a discrete term, for then one can infer a conclusion from
two affirmatives, e.g. it follows validly: 'every man is Socrates,
Plato is Socrates, therefore Plato is a man'. And such a
syllogism can be proved, because if the propositions are con-
verted there will result an expository syllogism in the third
figure.
The proof offered at the end of that text is evidently connected
with the Aristotelian ecthesis (13.13), as is suggested by the scholastic
term 'expository syllogism' and the following text from Ockham:
34.05 Besides the aforesaid syllogisms, there are also
expository syllogisms, about which we must now speak.
Where it is to be known that an expository syllogism is one
which is constituted by two singular premisses arranged in
the third figure, which, however, can yield both a singular,
and a particular or indefinite conclusion, but not a universal
one, just as two universals in the third figure cannot yield a
universal. ... To which it must be added that the minor
must be affirmative, because if the minor is negative the
syllogism is not valid. ... If the minor is affirmative, whether
the major is affirmative or negative, the syllogism is always
valid.
Stephen de Monte summarizes this doctrine in systematic
fashion :
34.06 But it is asked whether we can rightly syllogize by
means of an expository syllogism in every figure ; I say that we
can. For affirmatives hold in virtue of this principle : when two
different terms are united with some singular term taken
singularly and univocally, in some affirmative copulative
proposition from which the consequence holds to two uni-
versal affirmatives (de omni), such terms should be mutually
united in the conclusion. . . . But negatives hold in virtue of
this principle: whenever one of two terms is united with a
singular term etc., truly and affirmatively, and the other
negatively, such terms should be mutually united negatively,
respect being had to the logical properties. . . .
Seven syllogisms arise in this way, two in each of the first and
third figures, three in the second.
SCHOLASTIC LOGIC
B. ANALYSIS OF 'EVERY' AND 'SOME'
34.07 We proceed to the signs which render (propositions)
universal or particular. ... Of such signs, one is the universal,
the other is the particular. The universal sign is that by which
it is signified that the universal term to which it is adjoined
stands copulatively for its suppositum (per modum copula-
tionis). . . . The particular sign is that by which it is signified
that a universal term stands disjunctively for all its supposita.
And I purposely say 'copulatively' when speaking of the uni-
versal sign, since if one says : 'every man runs 'it follows formal-
ly: 'therefore this man runs, and that man runs, etc' But of
the particular sign I have said that it signifies that a universal
term to which it is adjoined stands disjunctively for all its
supposita. That is evident since if one says: 'some man runs'
it follows that Socrates or Plato runs, or Cicero runs, and so
of each (de singulis). This would not be so if this term did
not stand for all these (supposita); but it is true that this is
disjunctive. Hence it is requisite and necessary for the truth
of this: 'some man runs', that it be true of some (definite)
man to say that he runs, i.e. that one of the singular (pro-
positions) is true which is a part of the disjunctive (proposi-
tion) : 'Socrates (runs) or Plato runs, and so of each', since it is
sufficient for the truth of a disjunctive that one of its parts be
true (cf. 31.10 and 31.223).
This is the quite 'modern' analysis of quantified propositions
(44.03) in the following equivalences:
34.071 (For all x: x is F) if and only if: [a is F) and (b is F)
and (c is F) etc.
34.072 (There is an x such that x is F) if and only if:
{a is F) or (b is F) or (c is F) etc.
Further remarkable is the express appeal to a propositional rule.
In this text propositional logic is consciously made the basis of term-
logic, and this is only one of many examples.
C. EXPONIBLE PROPOSITIONS
The so-called'exponible' propositions were scholastically discussed
in considerable detail. They are those which are equivalent to a
product or sum of a number of categoricals. There are three kinds,
the exclusive, the exceptive, and the reduplicative. In view of the
metalogical treatment we give the 'logical squares' of Tarteret for the
first two kinds, with a substitution, also from him, and the mnemonic
expressions:
234
OTHER FORMULAE
34.08 DIVES
'Only man is an
animal' is thus
expounded : CONTRARY
(1) man is an
animal and (2)
nothing which is
not man is an
animal.
OR AT
'Only man is
not an animal' :
(1) man is not
an animal and
(2) everything
that is not man
is an animal.
(X
w
H
<
pq
D
C/2
%
■I
-•\
^
CT
-O
o
*
>
CO
CO
>
r
H
M
'Not only man is
not an animal' :
(1) every man is
an animal or (2)
something which
is not a man is
not an animal
ANNO
SUB-
CONTRARY
'Not only man
is an animal' :
(1) man is not
an animal or
(2) something
which is not
man is an animal.
HELI
34.09 AM ATE
'Every man besides
Socrates runs' :
PECCA TA
'Every man besides
Socrates does not
run':
(1) every man who (1) every man who
is not Socrates CONTRARY is not Socrates does
runs and (2) not run and (2)
Socrates is a man Socrates is a man
and (3) Socrates and (3) Socrates
does not run. runs.
235
SCHOLASTIC LOGIC
O 4y
'Not every man SUB- 'Not every man
besides Socrates CONTRARY besides Socrates
does not run' :
runs' : (1) some
(1) some man who
man who is not
is not Socrates
Socrates does not
runs, or (2) Socra-
run or (2) Socrates
tes is not a man or
is not a man or
(3) Socrates does
Socrates runs.
not run.
IDOLES
COM MOD.
The originality of the formal laws by substitution in which the
consequences shown in these squares are gained, consists in their
being a combination of the theory of consequences (especially the
so-called 'de Morgan' laws, cf. 31.35 f.) with the Aristotelian doctrine
of opposition (logical square: 12.09 f.). They are all valid, and one
can only marvel at the acumen of those logicians who knew how
to deduce them without the aid of a formalized theory. How com-
plicated are the processes of thought underlying the given schemata
can be shown by one of the simplest examples, in which ANNO
follows from D IVES. DIVES must be interpreted thus :
(1) Some M is L, and: no not-M is L. From that there follows by
the rule 31.222:
(2) no not-M is L,
and from that in turn, by the law of subalternation (24.29, cf. 32.44) :
(3) some not-M is not L.
Applying the rule 31.10 (cf. 31.223) one obtains :
(4) every M is L or some not-M is not L which was to be proved.
D. OBLIQUE SYLLOGISMS
The Aristotelian moods with 'indirect' premisses (16.24ff.) were
also systematically elaborated and developed by the Scholastics.
Ockham (34.10) already knew more than a dozen formulae of this
kind. But so far as we know, no essentially new range of problems
236
ANTINOMIES
was opened up. We cite some substitutions in such moods from
Ockham; their discovery has been quite groundlessly attributed to
Jungius.
34.11 It also follows validly: 'every man is an animal,
Socrates sees a man, therefore Socrates sees an animal'.
34.12 It follows validly: 'every man is an animal, an ass
sees a man, therefore an ass sees an animal'.
34.13 It follows: 'no ass belongs to man, every ass is an
animal, therefore some animal does not belong to man'.
§35. ANTINOMIES
A. DEVELOPMENT
Concerning also the search for solutions of antinomies in the Middle
Ages insufficient knowledge is available for us to be able to survey
the whole development here, though J. Salamucha devoted a serious
paper to it (35.01). The connected problems seem to have been well
known in the middle of the 13th century, but without their impor-
tance being realized. Albert the Great merely repeats the Aristotelian
solution of the Liar (35.02), and again Giles of Rome (in the second
half of the 13th century) only treats this antinomy briefly and quite
in the Aristotelian way (35.03). Peter of Spain, whose Summulae
treat of all the problems then considered important, considers the
fallacy of what is 'simply and in a certain respect' (under which
heading Aristotle deals with the Liar, cf. 23.18) (35.04), but says
nothing about antinomies.
However, two points are worth noting about Albert the Great; he
is the first that we find using the expression 'insoluble' (insolubile)
which later became a technical term in this matter, and then he has
some formulations that are new, at least in detail. This can be seen
in a passage from his Elenchics :
35.05 I call 'insoluble' those (propositions) which are so
formed that whichever side of the contradictory is granted,
the opposite follows. . . e.g. someone swears that he swears
falsely; he swears either what is true, or not. If he swears that
he swears falsely, and swears what is true, viz. that he swears
falsely, nobody swears falsely in swearing what is true : there-
fore he does not swear falsely, but it was granted that he
does swear falsely. But if he does not swear falsely and swears
that he swears falsely, he does not swear what is true ; there-
237
SCHOLASTIC LOGIC
fore he swears falsely: because otherwise he would not swear
what is true when he swears that he swears falsely.
By the time of Pseudo-Scotus the subject has become a burning
one; he cites at least one solution that diverges from his own (35.06)
and treats the question in two chapters of which the first bears the
title 'Whether a universal term can stand for the whole proposition
of which it is a part' (35.07). The answer is a decisive negative :
35.08 It is to be said that a part as part cannot stand for
the whole proposition.
His solution, however, does not consist in an application of this
principle, but is found in the distinction between the signified and the
exercised act:
35.09 If it is said: 'I say what is false, therefore it is true
that I say what is false', I answer that the consequence does
not hold formally, as (also) it does not follow: 'man is an ani-
mal, therefore it is true to say that man is an animal', although
the consequent is contained in the antecedent in the exercised
act. Granted further that it follows, though not formally, I
say that this other does not follow: 'I say that I say what is
false, therefore in what I say I am simply truthful', or only
in a certain respect and not simply. . . . Similarly it follows in
some cases: 'What I say is true, therefore I am simply truth-
ful', as (e.g.) here : 'It is true that I say that man is an animal,
therefore I am simply truthful', viz. in those cases in which
there is truth both in the act signified and the act exercised.
But in our case (in proposito) there is falsity in the act signified
and truth in the act exercised. It follows then : 'It is true that I
exercise the act of speaking about what is false; therefore that
about which I exercise it is false'.
A comparison of this text with 27.13 shows that we have here
almost exactly the modern distinction between use and mention. But
Pseudo-Scotus, employing the same terminology, teaches just the
opposite to Burleigh.
These two examples are enough to show the state of affairs in the
13th century. When we come to Ockham the antinomies are no
longer dealt with in sophistics, but in a special chapter About
Insolubles (35.10). After that such a treatise becomes an essential
part of scholastic logic. We pass over the further stages of develop-
ment, which are mostly not known, and show how far the matter
had got by the time of Paul of Venice at the end of the Middle Ages.
238
ANTINOMIES
B. FORMULATION
1. The Liar
35.11 I compose the much-disputed insoluble by positing
(1) that Socrates utters this proposition: 'Socrates says what
is false', and this proposition is A, and (2) (that he) utters no
other (proposition besides A), (where the proposition A) (3)
signifies so exactly and adequately that it must not be varied
in the present reply. That posited, I submits and ask whether
it is true or false. If it is said that it is true, contrariwise : it is
consistent with the whole case that there is no other Socrates
but this Socrates, and that posited, it follows that A is false.
But if it is said that A is false, contrariwise: it is consistent
with the whole case that there are two Socrateses of which
the first says A, and the second that there is no God: if that
is taken with the statement of the case, it follows that A is true.
35.12 I suppose therefore that Socrates, who is every
Socrates, utters this and no other proposition: 'Socrates says
what is false', which exactly and adequately signifies (what it
says) ; let it be A. Which being supposed, it follows from what
has been said that A is false; and Socrates says A, therefore
Socrates says what is false. This consequence is valid, and the
antecedent is true, therefore also the consequent; but the
antecedent is A, therefore A is true.
Secondly it is argued: What is false is said by Socrates,
therefore Socrates says what is false. The consequence holds
from the passive to its active. But the antecedent is true,
therefore also the consequent, and the antecedent is A,
therefore A is true. Since, however, the antecedent is true, it is
evident that its adequate significate is true. But it is a con-
tradiction that it should be true.
Thirdly it is argued: the contradictory opposite of A is
false, therefore A is true. The consequence holds and the
antecedent is proved: for this: 'no Socrates says what is
false' is false, and this is the contradictory opposite of A ;
therefore the contradictory opposite of A is false. The conse-
quence and the minor premiss hold, and I prove the major:
Since, A is false; but a certain Socrates says A; therefore a
certain Socrates says what is false. Or thus: No Socrates says
what is false; therefore no Socrates says the false .4.. The
consequence holds from the negative distributed super-
ordinate to its subordinate. The consequent is false, therefore
also the antecedent.
239
SCHOLASTIC LOGIC
2. Other antinomies
Besides this 'famous' insoluble there is a long series of similar
antinomies that derive from it, of which we give some examples
from Paul of Venice, omitting the always recurring words 'Socrates
who is all Socrates' and 'which signify exactly as the terms suggest
(pretendunt)' :
35.13 Socrates . . . believes this: 'Socrates is deceived' . . .
and no other (proposition).
35.14 Socrates believes this and no other: 'Plato is deceiv-
ed' .. . but Plato . . . believes this: 'Socrates is not deceived'.
35.15 Socrates . . . says this and nothing else: 'Socrates
lies'.
35.16 'Socrates is sick'; 'Plato answers falsely (male)';
'Socrates will have no penny'; ('Socrates will not cross the
bridge') ;* where it is supposed that every sick man, and only
one such, says what is false, and that every well man, and
only one such, says what is true (and correspondingly for the
three other cases). . . . On these suppositions I assert that
Socrates . . . utters only the following: 'Socrates is sick' etc.
Those are the so-called 'singular insolubles'. There follow on them
the 'quantified' ones:
35.17 I posit the case that this proposition 'it is false' is
every proposition.
35.18 Let this be the case, that there are only two proposi-
tions, A and B, A false, and B this: 'A is all that is true'.
35.19 I posit that A, B and C are all the propositions,
where A and B are true, and C is this: 'every proposition is
unlike this' indicating A and B.
35.20 I posit that A and B are all the propositions, where
A is this : 'the chimera exists' . . . and B this : 'every proposi-
tion is false'.
35.21 Let A, B and C be all the propositions . . . where A is
this: 'God exists', Bthis: 'man is an ass', C** this: 'there are
as many true as false propositions'.
35.22 The answer to be given would be similar on the
supposition that there were only five propositions ... of
which two were true, two false, and the fifth was: 'there are
more false than true (propositions)'.
* Inserted according to the words just following.
* * The text has D.
240
ANTINOMIES
Then some 'exponible' insolubles:
35.23 I posit that 'this is the only exclusive proposition'
is the only exclusive (proposition). . . .
35.24 Let this be a fallacy about exceptives: 'no proposition
besides A is false', supposing that this is A, and that it is
every proposition.
35.25 I posit that A, B, and C are all the propositions . . .
that A and B are true, and that C is this exclusive: 'every
proposition besides the exclusive is true'.
35.26 The answer is similar ... on the supposition that
every man besides Socrates says: 'God exists', and that
Socrates says only this: 'every man besides me says what is
true'.
These are only a few examples from the rich store of late scholastic
sophistic.
C. SOLUTIONS
1. The first twelve solutions
35.27 The first opinion states that the insoluble is to be
solved by reference to the fallacy of the form of speech (11.19).
. . . And if it is argued: 'Socrates utters this falsehood, there-
fore Socrates says what is false', one denies the consequence
and says: 'This is the fallacy of the form of speech, because
by reason of the (reference of the) speech the term 'false'
supposes for 'Socrates etc' in the antecedent, but for some-
thing else in the consequent. . . .
35.28 The second opinion solves the insolubles by the
fallacy of false cause (11.24) . . . since the antecedent seems
to be the cause of the consequent but is not. . . .
35.29 The third opinion says that when Socrates says
'Socrates says what is false', the word 'says', although in the
present tense, ought to be understood for the time of the
instant immediately preceding the time of utterance. There-
fore it denies it (the proposition), saying that it is false. And
then to the argument: 'this is false and Socrates says it,
therefore Socrates says a false (proposition)', they say that
the verb 'says' is verified for different times in the antecedent
and consequent. . . .
35.30 The fourth opinion states that nobody can say that he
says what is false or understand that he understands what is
false, nor can there be any proposition on which an insoluble
can be based. This opinion is repugnant to sense and thought.
241
SCHOLASTIC LOGIC
For everybody knows that a man can open his mouth and
form these utterances: 'I say what is false' or sit down and
read similar ones. . . .
35.31 The fifth opinion states that when Socrates says
that he himself says what is false, he says nothing. . . . This
opinion is likewise false because in so saying, Socrates says
letters, syllables, dictions and orations as I have elsewhere
shown. Further Socrates is heard to speak, therefore he says
something. Again they would have to say that if this, and no
other, proposition was written : 'it is false', that nothing would
be written, which is evidently impossible.
The fifth opinion counts the insoluble as deprived of sense.
35.32 The sixth opinion states that the insoluble is neither
true nor false but something intermediate, indifferent to
each. They are wrong too, because every proposition is true
or false, and every insoluble is a proposition, therefore every
insoluble is true or false. . . .
That is an effort to solve the antinomy in a three-valued logic.
35.33 The seventh opinion states that the insoluble is to
be solved by reference to the fallacy of equivocation. For when
it is said: 'Socrates says what is false' they distinguish about
the 'saying' according to an equivocation: for it can signify
saying that is exercised or that is thought (conceptum). And
by 'saying that is exercised' is meant that which is in course of
accomplishment; it expresses the judgment and is not com-
pletely a dictum. But by 'saying that is thought' is meant
(what happens) when a man has said something or spoken in
some way and immediately after he says that he says that,
or speaks in that way. E.g. supposes that Socrates says
'God exists' and immediately after: 'Socrates says what is
true'. This opinion says that when Socrates begins to say
'Socrates says what is false', if 'saying' be taken for exercised
saying, it is true; but if for saying in thought, it is false. And
if it is argued: 'nothing false is said by Socrates; and this is
said by Socrates; therefore it is not false' - they say that the
major is verified for saying in thought, and the minor for
exercised saying, and so (the argument) does not conclude.
But this solution is no use, for let it be supposed that the
speech is made with exercised saying, and the usual deduction
will go through. . . .
242
ANTINOMIES
This solution corresponds with that of Pseudo-Scotus above
(35.09).
35.34 The eighth opinion states that no insoluble is true or
false because nothing such is a proposition. For although
every or any insoluble be an indicative statement signifying
according as its signification is or is not, yet this is not
sufficient for it to be called a 'proposition'. Against this
opinion it is argued that it follows from it that there are some
two enunciations of which the adequate significate is one and
the same, yet one is a proposition, the other not, as is clear
when one supposes these: 'this is false' and 'this is false',
indicating in both cases the second of them. . . .
This is again a quite 'modern' conception. Paul of Venice, and. it
would seem, the majority of late Scholastics, did not like it.
35.35 The ninth opinion states that the insoluble is true or
false, but not true and not false. . . .
Here the alternative lA is true or false' seems to be admitted, but
'A is true' and 'A is false' to be both rejected.
35.36 The tenth opinion solves the insoluble by reference
to the fallacy of in a certain respect and simply (11.24),
saying that an insoluble is a difficult paradox (paralogismus)
arising from (a confusion between what is) in a certain respect
and simply, due to the reflection of some act upon itself with
a privative or negative qualification. So in solving, it says that
this consequence is not valid : 'this false thing is said by
Socrates, therefore a false thing is said by Socrates', supposing
that Socrates says the consequent and not something else
which is not part of it - because the argument proceeds from
a certain respect to what is simply so ; for the antecedent only
signifies categorically, but the consequent hypothetically,
since it signifies that it is true and that it is false. . . .
35.37 The eleventh opinion, favouring the opinion just
expounded, states that every insoluble proposition signifies
that it is true and that it is false, when understood as referring
to its adequate significate. For, as is said, every categorical
proposition signifies that that for which the subject and
predicate suppose is or is not the same thing, and the being or
not being the same thing is for the proposition, affirmative or
negative, to be true; therefore every categorical proposition,
whether affirmative or negative, signifies that itself is true,
243
SCHOLASTIC LOGIC
and every insoluble proposition falsifies itself; therefore every
insoluble proposition signifies that it is true or that it is
false. . . .
The last two opinions consider the insoluble to be equivalent to a
copulative proposition. Why it should be so we shall see below (35.44) .
35.38 The twelfth opinion, commonly held by all today, is
that an insoluble proposition is a proposition which is supposed
to be mentioned, and which, when it signifies precisely accord-
ing to the circumstances supposed, yields the result that it is
true and that it is false. E.g. if a case be posited about an
insoluble, and it is not posited how that insoluble should
signify, it is to be answered as though outside time: e.g. if
it be supposed that Socrates says: 'Socrates says what is
false' without further determination, the proposition advanc-
ed: 'Socrates says what is false' is to be doubted. But if it
be supposed that the insoluble signifies as the terms suggest,
the supposition is admitted and the insoluble is granted, and
one says that it is false. And if it be said : 'this is false : "Socra-
tes says what is false", therefore it signifies as it is not, but
signifies that Socrates says what is false, therefore etc' - the
consequence is denied. But in the minor it should be added
that it signifies precisely so, and if that is posited, every
such supposition is denied. . . .
The 'time of obligation' here referred to is a technical term of
scholastic discussion (tractatus de obligationibus: cf. § 26, D), on
which very little research has so far been done. It means the time
during which the disputant is bound to some (usually arbitrary)
supposition.
2. The thirteenth solution
35.39 The thirteenth opinion states a number of conjuncts,
some in the form of theses (conclusionum), others in the form
of suppositions, others in the form of propositions or corol-
laries ; but all these can be briefly stated in the form of theses
and corollaries.
35.40 The first thesis is this : no created thing can distinctly
represent itself formally, though it can do so objectively.
This is clear, since no created thing can be the proper and
distinct formal cognition of itself; for if something was to be
so, anything would be so, since there would be no more
reason in one case than in another. E.g. we say that the
244
ANTINOMIES
king's image signifies the king not formally but objectively,
while the mental concept which we have of the king signifies
the king not objectively but formally, because it is the formal
cognition of the king. But if it be said that it represents itself
distinctly, this will be objectively, by another concept
(noiilia) and not formally, by itself.
35.41 Second thesis: no mental proposition properly so-
called can signify that itself is true or that itself is false.
Proof: because otherwise it would follow that some proper
and distinct cognition would be a formal cognition of itself,
which is against the first thesis.
From this thesis it follows that the understanding cannot
form a universal mental proposition properly so-called which
signifies that every mental proposition is false, such as this
mental (proposition): 'every mental proposition is false',
understanding the subject to suppose for itself; nor can it
form any mental proposition properly so-called which signifies
that any other is false which in turn signifies that the one
indicated by the first is false; nor any mental proposition
properly so-called which signifies that its contradictory is
true, as this one: 'this is true' indicating its contradictory. . . .
The last two texts contain a notably acute formulation of the
veto on circulus vitiosus (48.21), and so of the most important modern
idea about the solution of the antinomies.
35.42 The third thesis is this : a part of a mental proposition
properly so-called cannot suppose for that same proposition
of which it is a part, nor for the contradictory of that proposi-
tion; nor can a part of a proposition that signifies in an
arbitrary way suppose for the corresponding mental proposi-
tion properly so-called. From which it follows that if this
mental proposition is formed, and no other: 'every mental
proposition is universal', it would be false.
35.43 Fourth thesis: there might be a vocal or written or
mental proposition improperly so-called which had reflection
on itself, because all such signify in an arbitrary way and not
naturally, objectively but not formally. But a mental proposi-
tion properly so-called is a sign that represents naturally and
formally, and it is not in our power that such a sign should
signify whatever we want, as it is in the case of a vocal,
written or mental sign improperly so-called.
From this thesis it follows that every insoluble proposition
245
SCHOLASTIC LOGIC
is a vocal, written or mental proposition improperly so-called;
and a part of any such can suppose for the whole of which it
is a part.
35.44 The fifth thesis is this : to every insoluble proposition
there corresponds a true mental proposition properly so-
called, and another one properly so-called, false. This is
evident in the following: 'this is false' indicating itself, which
corresponds to one such mental proposition, 'this is false',
which is true. And the second part is proved. For this vocal
proposition is false, therefore it signifies that a mental one is
false, but not the one expressed, therefore another one which
is true, viz. 'this is false', indicating the first mental one which
indicates a vocal or written one.
35.45 From this thesis there follow some corollaries.
First, that any insoluble proposition, and its contradictory too,
is a manifold proposition (propositio plures) because there
correspond to it a number of distinct (inconiundae) mental
propositions.
Second, there are some propositions, vocally quite similar
and with terms supposing for the same things, one of which
is a manifold proposition, but not the other. This is clear in the
following: 'this is false' and 'this is false' where each 'this'
indicates the second proposition.
Third corollary, every insoluble proposition is simultaneous-
ly true and false, and its contradictory likewise, because two
mental propositions of which one is true and the other false
contradict one another, though neither is simply true or
simply false, but in a certain respect. . . .
3. The fourteenth solution
35.46 The fourteenth opinion, which is the basis of many
of the preceding ones and so of those disputants who try
rather to evade (the difficulties) than to answer, states that
the insolubles are to be solved by means of the fallacy of the
accident, according to which paradoxes (paralogismi) arise
in two ways, by variation of the middle term or of one of the
extremes. By variation of the middle, as when the middle
supposes for something different in the major to what it
supposes for in the minor, and conversely. And similarly
when an extreme is varied. This opinion therefore says that
when Socrates says 'Socrates says what is false', he says what
is false. And then in reply to the argument: 'Socrates says
246
ANTINOMIES
this, and this is false, therefore Socrates says what is false'
they deny the consequence, saying that here is a fallacy of the
accident due to variation in an extreme; for the term 'false'
supposes for something in the minor for which it does not
suppose in the conclusion. Similarly if it is argued from the
opposite saying of Socrates: 'nothing false is said by Socrates;
this is false; therefore this is not said by Socrates', this is a
fallacy of the accident due to variation of the middle ; for the
term 'false' supposes for something in the major for which it
does not suppose in the minor.
To show that, they presuppose that in no proposition does
a part suppose for the whole of which it is a part, nor is it
convertible with the whole, nor antecedent to the whole.
From which it is clear that the proposition 'Socrates says
what is false, signifies that Socrates says what is false, not,
however, the false thing that he says, but some false thing
distinct from that; but because he only says that proposition,
therefore it is false. . . .
This opinion has been met with already in Pseudo-Scotus (35.06;
and is adopted by others too.
4. Preliminaries to the solution of Paul of Venice
After expounding fourteen opinions none of which are acceptable
to him, Paul of Venice gives his own solution, and takes occasion to
collect the current late-scholastic teachings relevant to the antino-
mies. We reproduce the essentials.
35.47 To explain the fifteenth opinion, which I know to
be that of good (logicians) of old times, three chapters (articuli)
are adduced. The first contains an explanation of terms, the
second introductory suppositions, the third our purpose in the
form of theses.
35.48 As to the first, this is the first division : every insoluble
arises either from our activity or from a property of the
expression (vocis). Our acts are twofold, some interior, others
exterior. Interior are such as imagining, thinking etc. ; exterior
are bodily ones such as saying, speaking etc. Insolubles arising
from our activity are: 'Socrates says what is false', 'Socrates
understands what is false' etc. Properties of the expression
are such as being subject, having appellation, being true or
false, being able to be true, not being true of something other
than itself, and so simply (de se) false, and not being false of
itself or of something else. And so there arise from properties
247
SCHOLASTIC LOGIC
of the expression insolubles like these: 'it is false', 'nothing
is true', 'the proposition is not verified of itself. . .
35.49 The second division is this: some propositions have
reflection on themselves, some do not. A proposition having
reflection on itself is one whose signification reflects on itself,
e.g. 'it is every complex thing', or 'this is false', indicating
itself. A proposition without reflection on itself is one whose
significate is not referred to itself, e.g. 'God exists' and 'man
is an ass'.
35.50 The third division is this: of propositions having
reflection on themselves some have this reflection immediately,
others have it mediately. . . .
35.51 The fourth division: of propositions having reflec-
tions on themselves, some have the property that their
significations terminate solely at themselves, e.g. 'this is
true', 'this is false', indicating themselves. But others have
the property that their significations terminate both at
themselves and at other things, e.g. 'every proposition is
true', 'every proposition is false'. For they do not only signify
that they alone are true or false, but that other propositions
distinct from them are so too. . . .
35.52 It follows that no proposition has reflection on itself
unless it contains a term that is appropriated to signify the
proposition, such as are the terms 'true', 'false', 'universal',
'particular', 'affirmative', 'negative', 'to be granted', 'to be
denied', 'to be doubted' and so on. But not every proposition
containing such a term has to have reflection on itself, as is
clear in these cases: 'it is false', when this is true, and again
'this is true' indicating 'God exists'; for such does not have
reflection on itself, but its signification is directed solely to
what is indicated etc. . . .
There again is the 'modern' notion of the vicious circle (48.21).
Here are a few more preliminaries :
35.53 The first introductory supposition is this: that that
proposition is true whose adequate significate is true, and if
its being true contains no contradiction. . . .
35.54 Second supposition: that proposition is said to be
false which falsifies itself or whose falsity arises not from the
terms but from its false adequate significate. From which it
follows that there is a false proposition with a true adequate
significate, as is clear in the following: 'that is false', indicating
248
ANTINOMIES
itself. That it is false is evident, since it states that it is false,
therefore it is false; and so its adequate significate is true, since
it is true that it (the proposition) is false. It follows that
every proposition which falsifies itself is false, and that not
every proposition which verifies itself is true ; since this :
'every proposition is true' verifies itself but is not true, as is
evident.
35.55 The third supposition is this: two propositions are
equivalent (invicem convertuntur) if their adequate significates
are identical. For let A and B be two such propositions having
the same adequate significate, and I argue thus: A and B
have all extremes the same, vocally and in writing, and in
thought, and similar copulas, and there is no indication
belonging to one which does not belong to the other; then
they are equivalent.
There follow some further preliminary suppositions taken from
the generally received teaching about supposition and consequence.
Finally this:
35.56 The last supposition is this: a part of a proposition
can stand for the whole of which it is a part, as also for
everything which belongs to it, without restriction, whether
in thought or in writing or in speech.
Thereby is rejected the thirteenth opinion (35.39 ff.), and with
it the modern principle according to which an insoluble is not a
proposition since it contains a part standing for the whole (48.12 f.).
This principle seems to be presupposed in various ways by the
fourth (35.30), fifth (35.31), eighth (35.34), tenth (35.36)," and
eleventh (35.37) opinions.
The rejection of the thirteenth opinion means that the current
modern distinction of language and meta-language was not adopted
by Paul of Venice for his own solution. But it is explicitly accepted
in the fifth thesis of the thirteenth opinion (35.44), more or less so
in some of the other opinions.
5. The solution of Paul of Venice
Paul's own solution is very like that of the eleventh (35.37) and
twelfth (35.38) opinions, and so we do not reproduce his long and
difficult text. It consists essentially in a sharp distinction between
the ordinary and 'exact and adequate' meaning of the insoluble
proposition, where 'exact and adequate' connotes:
(1) the semantic correlate, that to which it refers;
(2) that the proposition itself is true.
249
SCHOLASTIC LOGIC
This was said already in 35.57, though without the use of 'exact
and adequate', and with a universality that Paul does not approve.
We repeat the main ideas, as they underlie his own solution. Some
simplification will be effected, and formalization used. The first
thing is to set out the antinomy, for which four extralogical axioms
are employed :
(1)^4 signifies: A is false.
(2) If A signifies p, then A is true if and only if p.
(3) If A signifies p, then A is false if and only if not-p.
(4) A is false if and only if A is not true. (1) is the 'insoluble'
proposition itself, (2)-(4) are various formulations of the Aristotelian
definitions of truth and falsity (10.35). Substituting lA is false'
for 'p' in (2), we get by (1):
(5) A is true if and only if A is false, which with (4) gives :
(6) A is true if and only if A is not true, which in turn yields :
(7) A is not true,
and so by (4) :
(8) A is false.
But if we put lA is false' for 'p' in (3), we get:
(9) A is false if and only if A is not false, which immediately
yields:
(10) A is not false,
in contradiction to (8). Here then is a genuine antinomy.
But the antinomy does not emerge if we operate with the 'exact
and adequate' meaning instead of the simple one. (1) and (4) remain,
but the other two axioms take on these forms :
(2') If A signifies p, then A is true if and only if [(1) A is true, and
(2) p].
(3') If A signifies p, then A is false if and only if not [(1) A is true,
and (2) p], since as has been said, a proposition has 'exact and
adequate' signification when it signifies that it is itself true, and that
what it states is as it is stated to be.
The first part of the deduction now goes through analogously to
that given above, and we again reach:
(8') A is false.
But putting 'A is false' for 'p' in (3') gives:
(9') A is false if and only if not [(1) A is true, and (2) A is false],
to which we can apply the de Morgan laws (cf. 31.35) to get:
(10') A is false if and only if either (1) A is not true, or (2) A is not
false, i.e. in view of (4):
(IT) A is false if and only if either (1) A is false or (2) A is not
false.
As that alternation is logically true, being a substitution in the
law of excluded middle (cf. 31.35), the first part of the equivalence
must also be true, giving us:
(12') A is false
250
SUMMARY
which so far from being in contradiction to (8') is equiform with it.
The antinomy is solved.
So far as we know, the medieval logicians only treated of semantical,
not of logical antinomies. But the solutions contain all that is required
for those as well.
SUMMARY
In summary, we can make the following statements about medie-
val formal logic, in spite of our fragmentary knowledge:
1. Scholasticism created a quite new variety of formal logic. The
essential difference between this and the one we found among the
ancients, consists in its being an endeavour to abstract the laws and
rules of a living (Latin) language, regard had to the whole realm of
semantical and syntactical functions of signs.
2. This endeavour led to the codification of a far-reaching and
thorough semantics and syntax; semiotic problems hold the fore-
front of interest, and nearly all problems are treated in relation to
them.
3. Hence this logic is nearly entirely conducted in a meta-language
(§ 26, B) with a clear distinction between rules and laws. Most of the
theorems are thought of as rules and formulated descriptively.
4. The problem of logical form (§ 26, C) is posed and solved with
great acumen.
5. Problems of propositional logic and technique are investigated
as thoroughly and in as abstract a way as anywhere among the
Megarian-Stoics.
6. Asserloric term-logic consists here essentially in a re-interpreta-
tion and acute development of syllogistic. But there are also other
kinds of problem in evidence, such as that of plural quantification, of
the null class, perhaps of relation-logic, etc.
7. Modal logic, both of propositions and terms, became one of the
most important fields of investigation. Not only was the traditional
system analysed with amazing thoroughness, but quite new prob-
lems were posed and solved, especially in the propositional domain.
8. Finally the problem of semantical antinomies wras faced in really
enormous treatises. Numerous antinomies of this kind were posited,
and we have seen more than a dozen different solutions attempted.
Between them they contain nearly every essential feature of what we
know today on this subject.
So even in the present incomplete state of knowledge, we can
state with safety that in scholastic formal logic we are confronted
with a very original and very fine variety of logic.
251
PART IV
Transitional Period
§36. THE 'CLASSICAL' LOGIC
It is usual to put the close of the medieval period of history at the
end of the 15th century. Of course that does not mean that typically
scholastic ways of thought did not persist longer; indeed very im-
portant scholastic schools arose in the 16th and 17th centuries and
accomplished deep and original investigations — it is enough to
mention Cajetan and Vittoria. But there was no more research into
formal logic; at most we find summaries of earlier results.
Instead there slowly grew up something quite new, the so-called
'classical' logic. Within this extensive movement which held the
field in hundreds of books of logic for nearly four hundred years, one
can distinguish three different tendencies: (1) humanism (inclusive
of those later 17th century thinkers who were humanist in their
approach to logic); it is purely negative, a mere rejection of Scho-
lasticism; (2) 'classical' logic in the narrower sense; (3) more recent
endeavours to broaden the bounds of (2). Typical examples of the
three are L. Valla and Peter Ramus, the Logique du Port Royal,
and W. Hamilton.
In what follows we shall first quote some passages to illustrate
the general attitude of authors of books entitled 'Logic' in this
period, then some that contain contributions to logical questions,
whether scholastic or mathematico-logical, though these contribu-
tions are of small historical importance.
A. HUMANISM
Interest centres much more on rhetorical, psychological and
epistemological problems than on logical ones. The humanists, and
many 'classical' logicians after them, expressly reject all formalism.
That they did not at the same time reject logic entirely is due to
their superstitious reverence for all ancient thinkers, Aristotle in-
cluded. But everything medieval was looked on as sheer barbarism,
especially if connected with formal logic. Here is an instance. Valla
writes :
36.01 I am often in doubt about many authors of the dia-
lectical art, whether to accuse them of ignorance, vanity or
malice, or all at once. For when I consider the numerous
errors by which they have deceived themselves no less than
others, I ascribe them to negligence or human weakness. But
when on the other hand I see that everything they have
transmitted to us in endless books has been given in quite a
few rules, what other reason can I suppose than sheer pride?
In amusing themselves by letting the branches of the vine
254
H U M A N I S M
spread far and wide, they have changed the true vine into a
wild one. And when - this is the worst - I see the sophisms,
quibbles and misrepresentations which they use and teach,
I can only kindle against them as against people who teach the
art of piracy rather than navigation, or to express myself
more mildly, knowledge of wrestling instead of war.
And again, about the third figure of the syllogism:
36.02 0 trifling Polyphemus! 0 peripatetic family, that
loves trifles! 0 vile people, whoever have you heard arguing
like this? Indeed, who among you has ever presumed to
argue so? Who permitted, endured, understood one who
argued thus?
It is on these 'grounds' that the third figure is to be invalid!
In another manner, but the content goes deeper, Descartes ex-
presses himself:
36.03 We leave out of account all the prescription of the
dialecticians by which they think to rule human reason,
prescribing certain forms of discourse which conclude so
necessarily that in relying on them the reason, although to
some extent on holiday from the informative and attentive
consideration (of the object), can yet draw some certain
conclusion by means of the form.
Evidently such an attitude will not bring one to any logic. Peter
Ramus holds a special position among the humanists. Though, at
least in his first period, he was perhaps the most radical anti-
Aristotelian, yet he succeeded in formulating on occasion some
interesting thoughts, and published extensive treatises on formal
logic. However, the following gives some idea of the general level
of his logic:
36.04 Moreover, two further connected (i.e. conditional)
moods were added by Theophrastus and Eudemus, in which
the antecedent is negative and the consequent affirmative.
The third connected mood, then, takes the contradictory of
the antecedent and concludes to the contradictory (of the
consequent), e.g.
If the Trojans have come to Italy without due permission,
they will be punished;
but they came with permission,
therefore they will not be punished. . . .
255
TRANSITIONAL PERIOD
36.05 The fourth connected mood takes the consequent
and concludes to the antecedent: . . .
If nothing bad had happened, they would be here already;
but they are here,
therefore nothing bad has happened.
36.06 This mood is the rarest of all, but natural and useful,
strict and correct, and it never produces a false conclusion
from true premisses. ... *
Of course Theophrastus taught no such moods; both theorems
are formally invalid and hold only by reason of the matter in parti-
cular cases. It is instructive to compare these thoughts with the
treatment of similar problems in the Stoics (22.04 f.) and Scholastics
(§31).
B. CONTENT
In so bad a milieu logic could not last long, yet there were some
thinkers among the humanists, Melanchthon for instance, who without
being creative logicians, had a good knowledge of Aristotle. It was
through them that in the 17th century the form of logic developed
which we call the 'classical' in the narrower sense, partly among
the so-called Protestant Scholastics, partly in Cartesian circles.
Perhaps the most important representative work is the Logique
ou Varl de penser of P. Nicole and A. Arnault. We describe the
contents of this work, since they give the best survey of the problems
considered in 'classical' logic.
The book has four parts, about ideas, judgments, arguments, and
method. In the first part the Aristotelian categories (ch. 3) and
predicables (ch. 7) are briefly considered, along with some points of
semantics (Des idees des choses el des idees des signes, ch. 2), com-
prehension and extension (ch. 6). The other eleven chapters are
devoted to epistemological reflections.
The second part roughly corresponds to the content of Aristotle's
Hermeneia, and includes also considerations on definition and
division (ch. 15-16).
In the third part the authors expound categorical syllogistic,
apparently following Peter of Spain and so as a set of rules, but with
use of singular premisses in the manner of Ockham (34.01 f.). Four
figures are recognized, with nineteen moods (the subaltern moods
are missing). There follows a chapter on hypothetical syllogisms,
and (ch. 12) a theory of syllogismes conjonctifs (the Stoic compounds)
using formulas of term-logic, e.g.
* Admittedly these moods, as Professor A. Church has stated, do not appear
in all editions. Perhaps Ramus saw his own mistake.
256
PSYCHOLOGISM
36.07 If there is a God, one must love him:
But there is a God ;
Therefore one must love him.
Then there are some considerations about dialectical loci.
When we compare this with scholastic logic, the main things
missing are the doctrines of supposition, consequences, antinomies,
and modal logic. The main topics covered are those of the Categories,
Hermeneia, and the first seven chapters of the first hook of the
Prior Analytics, but the treatment is often scholastic rather than
Aristotelian, for which we instance the use of the mnemonics
Barbara, Celarenl etc. and the metalogical method of exposition.
The Logica Hamburgensis of J. Jungius (1635) is much better,
and richer in content; but it did not succeed in becoming established.
The Logique ou Vart de penser, also called the Port Royal Logic,
became the standard text-book, a kind of Summulae of 'classical'
logic. All other text-books mainly repeated its contents.
C. PSYCHOLOGISM
'Classical' logic is characterized not only by its poverty of con-
tent but also by its radical psychologism. Jungius provides a good
example:
36.08 1. Logic is the art of distinguishing truth from falsity
in the operations of our mind (mentis).
2. There are three operations of the mind : notion or concept,
enunciation, and dianoea or discourse.
3. Notion is the first operation of our mind, in which we
express something by an image; in other words a notion is
a simulacrum by which we represent things in the mind. . . .
5. Enunciation is the second operation of the mind, so
compounded of notions as to bring about truth or falsity.
E.g. these are true enunciations: the sun shines, man is a
biped, the oak is a tree. . . .
9. It is to be noted that a notion and the formation of a
notion, an enunciation and the effecting of an enunciation,
an argumentation and the construction of an argumentation
are one and the same.
This is admittedly an extreme case. But when one thinks that
the text is from Jungius, one of the best logicians of the 17th
century, one cannot but marvel at the extent to which the under-
standing of logic has disappeared. Even Boole will maintain much
this idea of logic.
257
TRANSITIONAL PERIOD
Poor in content, devoid of all deep problems, permeated with a
whole lot of non-logical philosophical ideas, psychologist in the
worst sense, - that is now we have to sum up the 'classical' logic:
It may, however, be remarked that A. Menne (Logik und Existenz,
131, note 34) has propounded a distinction between an at least
relatively pure 'classical' logic, and a 'traditional' philosophical and
psychological logic, though these terms are commonly used syno-
nymously. Of the former, J. N. Keynes (1906) may be taken as in
every sense the best representative, but W. E. Johnson (1921)
shows how relative is the distinction.
D. LEIBNIZ
Formed by this logic and its prejudices, modern philosophers such
as Spinoza, the British empiricists, Wolff, Kant, Hegel etc. could
have no interest for the historian of formal logic. When compared
with the logicians of the 4th century B.C., the 13th and 20th cen-
turies a.d. they were simply ignorant of what pertains to logic and
for the most part only knew what they found in the Port Royal
Logic.
But there is one exception, Leibniz (1646-1716). So far from being
an ignoramus, he was one of the greatest logicians of all time, which
is the more remarkable in that his historical knowledge was rather
limited. His place in the history of logic is unique. On the one hand
his achievement constitutes a peak in the treatment of a part of the
Aristotelian syllogistic, where he introduced many new, or newly
developed features, such as the completion of the combinatorial
method, the exact working out of various methods of reduction,
the method of substitution, the so-called 'Eulerian' diagrams, etc.
On the other hand he is the founder of mathematical logic.
The reason why Leibniz is, nevertheless, named in this section,
and only named, is that his great achievements in the realm of
mathematical logic are little relevant to the history of problems,
since they remained for long unpublished and were first discovered
at the end of the 19th century when the problems he had dealt
with had already been raised independently.
Only in one respect does he seem to have exercised a decisive
influence, in forming the idea of mathematical logic. The pertinent
passages will be quoted incidentally in the next section. Here we
limit ourselves to quoting some of his contributions to syllogistic
theory, and showing some of his diagrams.
E. COMPREHENSION AND EXTENSION
The idea underlying the distinction between comprehension and
extension is a very old one: it is presupposed, for instance, in the
Isagoge of Porphyry (24.02 ff.); the scholastic doctrine of supposition
258
SYLLOGISTIC
has a counterpart of it in the theory of simple (27.17 IT. ) and personal
(27.24 fT.) supposition with an elaborate terminology. But the
expressions comprehension and elendue are first found in the Port
Royal Logic. Leibniz evidently has the idea, but without an establish-
ed terminology.
We first cite an extract from his article De formae logicae com-
probatione per linearum ductus:
36.09 Up to now we have assessed the quantities of terms
in respect of (ex) the individuals. And when it was said:
'every man is an animal', it was meant (consider alum est)
that all human individuals form a part of the individuals
that fall under 'animal' (esse partem individuorum animalis).
But in respect of (secundum) ideas, the assessment proceeds
just conversely. For while men are a part of the animals,
conversely the notion of animal is a part of the notion apply-
ing to man, since man is a rational animal.
So Leibniz had a fairly accurate idea of comprehension and
extension, as well as of their inter-relationship. Now we come to the
Port Royal Logic:
36.10 Now in these universal ideas (idees) there are two
things (choses) which it is important to keep quite distinct:
comprehension and extension.
I call the comprehension of an idea the attributes which
it contains and which cannot be taken away from it without
destroying it; thus the comprehension of the idea of triangle
includes extension, figure, three lines, three angles, the
equality of these three angles to two right-angles, etc.
I call the extension of an idea the subjects to which it applies,
which are also called the inferiors of a universal term, that
being called superior to them. Thus the idea of triangle in
general extends to all different kinds of triangle.
F. THE FOURTH FIGURE AND SUBALTERN MOODS
Already in his youthful work De arte combinatoria Leibniz resumed
the thought of Albalag (32.25 ff.), without being acquainted with it,
and proved that there is a fourth figure of assertoric syllogism
(36.11). Later, he gave a complete and correct table of the twenty-
four syllogistic moods, in which he deduced the moods of the second
and third figures from those of the first, using the first reduction
procedure of Aristotle (§ 14, D). We give a table which reproduces
the deduction of the second and third figure moods, this time in the
original language. ' Regressus' means contraposition.
259
TRANSITIONAL PERIOD
36.12
Barbara primae
A CD
ABC
ABD
Barbara primae
ACD
ABC
ABD
Regressus
ACD
OBD
Regr.
ABC
OBD
Ergo
OBC
Ergo
OCD
Hinc Baroco
Hinc Bocardo
secundae
ACD
OBD
OBC
tertiae
OBD
ABC
OCD
Celarent primae
ECD
ABC
EBD
Celarent primae
ECD
ABC
EBD
Regr.
ECD
IBD
Regr.
ABC
IBD
Ergo
OBC
Ergo
ICD
Hinc Festino
Hinc Disamis
secundae
ECD
IBD
OBC
tertiae
IBD
ABC
ICD
Darii primae
ACD
IBC
IBD
Darii primae
ACD
IBC
IBD
Regr.
ACD
EBD
Regr.
IBC
EBD
Ergo
EBC
Ergo
OCD
Hinc Camestres
Hinc Ferison
secundae
ACD
EBD
EBC
tertiae
EBD
IBC
OCD
Ferio primae
ECD
IBC
OBD
Ferio primae
ECD
IBC
OBD
Regr.
ECD
ABD
Regr.
IBC
ABD
Ergo
EBC
Ergo
ICD
Hinc Cesare
Hinc Datisi
secundae
ECD
ABD
EBC
tertiae
ABD
IBC
ICD
Barbari primae
ACD
ABC
IBD
Barbari primae
ACD
ABC
IBD
Regr.
ACD
EBD
Regr.
ABC
EBD
Ergo
OBC
Ergo
OCD
Hinc Camestres
Hinc Felapton
secundae
ACD
EBD
OBC
tertiae
EBD
ABC
OCD
Celaro primae
ECD
ABC
OBD
Celaro primae
ECD
ABC
OBD
Regr.
ECD
ABD
Regr.
ABC
ABD
Ergo
OBC
Ergo
ICD
Hinc Cesaro
Hinc Darapti
secundae
ECD
ABD
OBC
tertiae
ABD
ABC
ICD
(cf. 13.21, 13.22.
G. SYLLOGISTIC DIAGRAMS
The idea of representing class relations and syllogistic moods by
geometrical figures was familiar to the ancient commentator
(24.34) ; how far it was current among the Scholastics is not yet
known. The use of circles is usually ascribed to L. Euler (1701-83)
(cf. his Lettres a une princesse d'Allemagne, 1768), while that of
straight lines is associated with the name of Lambert. But the
former are to be found earlier in J. C. Sturm (1661) (36.13), and
the latter in Alstedius (1614) (36.14). Schroder (36.15) notes that
L. Vives was using angles and triangles in 1555.* Leibniz's use
For the foregoing and some further details vid. A. Menne (36.16)
260
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36.18
VENN-DIAGHAMS
Y Z
•261
QUANTIFICATION
of circles and other diagrammatic methods remained unpublished
till 1903. We reproduce a page of his MS (36.17) which contains both
circular and rectilinear diagrams.
Such methods of presentation were much considered and further
developed from the time of Euler onwards. J. Venn (1860) intro-
duced ellipses for his investigations of the relations between more
than three classes, and marked with a star every region representing
a non-void class. Three of his diagrams are reproduced in (36.18)
and a systematic development is considered by W. E. Hocking
(36.19).
A different kind of diagram, mnemonic rather than expository
of probative, due to Johnson, may be added here.
36.20 The attached diagram, taking the place of the mnemo-
nic verses, indicates which moods are valid, and which are
common to different figures. The squares are so arranged that
the rules for the first, second and third figures also show the
compartments into which each mood is to be placed, according
as its major, minor or conclusion is universal or particular,
affirmative or negative. The valid moods of the fourth figure
occupy the central horizontal line.
In the figure, the superscripts V and V indicate the propositions
that may be weakened or strengthened by subalternation.
H. QUANTIFICATION OF THE PREDICATE
While all points so far referred to fall within the general scheme
of Aristotelian logic, Bentham's doctrine of the quantification of
the predicate, usually ascribed to Hamilton, is directly opposed to
Aristotle's teaching (12.03). At the same time, as can be seen from
the texts, it is a development of the scholastic doctrine of exponibles.
It has this historical importance, that it shows the kind of problem
being considered by logicians at the time of Boole, and in some
degree throws light on the origin of Boole's calculus.
262
TRANSITIONAL PERIOD
We give two texts from G. Bentham (1827) first, then one from
Hamilton (1860):
36.21 In the case where both terms of a proposition are
collective entities, identity and diversity may have place:
1. Between any individual referred to by one term, and any
individual referred to by the other. Ex. : The identity between
equiangular and equilateral triangles.
2. Between any individual referred to by one term, and
any one of a part only of the individuals referred to by the
other. Ex. : The identity between men and animals.
3. Between any one of a pari only of the individuals referred
to by one, and any one of a pari only of the individuals referred
to by the other term. Ex. : The identity between quadrupeds
and swimming animals.
36.22 Simple propositions, considered in regard to the above
relations, may therefore be either affirmative or negative;
and each term may be either universal or partial. These
propositions are therefore reducible to the eight following
forms, in which, in order to abstract every idea not connected
with the substance of each species, I have expressed the two
terms by the letters X and Y, their identity by the mathe-
matical sign =, diversity by the sign || , universality by the
words in toto, and partiality by the words ex parte; or, for
the sake of still further brevity, by prefixing the letters I and
p, as signs of universality and partiality. These forms are,
1.
X in toto =
Y ex parte
or IX
= pY
2.
X in toto
Y ex parte
or IX
II PY
3.
X in toto =
Y in toto
or IX
= IY
4.
X in toto
Y in toto
or IX
1 IF
5.
X ex parte =
Y ex parte
or pX
= PY
6.
X ex parte
Y ex parte
or pX
1 PY
7.
X ex parte =
Y in toto
or pX
= tY
8.
X ex parte \\
Y in toto
or pX
II tY
Hamilton writes:
36.23 The second cardinal error of the logicians is the not
considering that the predicate has always a quantity in
thought, as much as the subject; although this quantity be
frequently not explicitly enounced, as unnecessary in the
common employment of language ; for the determining notion
or predicate being always thought as at least adequate to, or
263
QUANTIFICATION
coextensive with, the subject or determined notion, it is
seldom necessary to express this, and language tends ever
to elide what may safely be omitted. But this necessity
recurs, the moment that, by conversion, the predicate becomes
the subject of the proposition; and to omit its formal state-
ment is to degrade Logic from the science of the necessities
of thought, to an idle subsidiary of the ambiguities of speech.
An unbiased consideration of the subject will, I am confident,
convince you that this view is correct.
1°, That the predicate is as extensive as the subject is
easily shown. Take the proposition, - 'All animal is man', or,
'All animals are men'. This we are conscious is absurd. . . .
We feel it to be equally absurd as if we said, - 'All man is all
animal', or ,'A11 men are all animals'. Here we are aware that
the subject and predicate cannot be made coextensive. If we
would get rid of the absurdity, we must bring the two notions
into coextension, by restricting the wider. If we say - 'Man
is animal', [Homo est animal], we think, though we do not
overtly enounce it, 'All man is animal'. And what do we mean
here by animal? We do not think, all, but some, animal. And
then we can make this indifferently either subject or predi-
cate. We can think, - we can say, 'Some animal is man',
that is, some or all man; and, e converso, 'Man (some or all) is
animal', viz. some animal. . . .
2°, But, in fact, ordinary language quantifies the predicate
so often as this determination becomes of the smallest import.
This it does either directly, by adding all, some, or their
equivalent predesignations to the predicate ; or it accomplishes
the same end indirectly, in an exceptive or limitative form.
Hamilton then proceeds to repeat, in dependence on the works
of various 17th and 18th century logicians, the scholastic doctrine
of the exponibilia (§ 34, C).
264
PART V
The mathematical variety of Logic
MATHEMATICAL LOGIC
I. General Foundations
§37. INTRODUCTION TO MATHEMATICAL LOGIC
A. CHARACTERISTICS
The development of the mathematical variety of logic is not yet
complete, and discussions still go on about its characteristic scope
and even about its name. It was simultaneously called 'mathemati-
cal logic', 'symbolic logic' and 'logistic' by L. Couturat, Itelson and
Lalande in 1901, and is sometimes simply called 'theoretical logic'.
Even apart from the philosophical discussions as to whether or
how far it is distinct from mathematics, there is no unanimity about
the specific characteristics which distinguish it from other forms of
logic.
However, there exists a class of writings which are generally
recognized as pertaining to 'mathematical logic' ('logistic', 'symbolic
logic' etc.). Analysis of their contents shows that they are predomi-
nantly distinguished from all other varieties of logic by two inter-
dependent characteristics.
(1) First, a calculus, i.e. a formalistic method, is always in evi-
dence, consisting essentially in the fact that the rules of operation
refer to the shape and not the sense of the symbols, just as in mathe-
matics. Of course formalism had already been employed at times
in other varieties of logic, in Scholasticism especially, but it is now
erected into a general principle of logical method.
(2) Connected with that is a deeper and more revolutionary
innovation. All the other varieties of logic known to us make use
of an abslractiue method ; the logical theorems are gained by abstrac-
tion from ordinary language. Mathematical logicians proceed in
just the opposite way, first constructing purely formal systems, and
later looking for an interpretation in every-day speech. This process
is not indeed always quite purely applied; and it would not be
impossible to find something corresponding to it elsewhere. But at
least since Boole, the principle of such construction is consciously
and openly laid down, and holds sway throughout the realm of
mathematical logic.
Those are the essential features of mathematical logic. Two more
should be added:
(3) The laws are formulated in an artificial language, and consist
of symbols which resemble those of mathematics (in the narrower
sense). The new feature here is that even the constants are expressed
in artificial symbols; variables, as we have seen, have been in use
since the time of Aristotle.
(4) Finally, until about 1930 mathematical logic formulated its
266
CHRONOLOGY
theorems in an object language, in this unlike the Scholastics, but
in conformity with the ancients. That this is no essential feature
is shown by more recent developments and the spread of metalogical
formulation. But till 1930 the use of the object language is charac-
teristic.
It may be further remarked that it can be said of mathematical
logic, what was finally said about scholastic, that it is very rich
and very formalistic. In wealth of formulae indeed, it seems to
exceed all other forms of logic. It is also purely formal, being sharply
distinguished from the decadent 'classical' logic by its avoidance of
psychological, epistemological and metaphysical questions.
B. CHRONOLOGICAL SEQUENCE
G. W. Leibniz generally ranks as the original mathematical
logician, but if he cannot count as the founder of mathematical
logic it is because his logical works were for the most part published
long after his death (the essentials by L. Couturat in 1901). However,
he had some successors, the most important of whom were the
brothers Bernoulli (1685), G. Plouquet (1763, 1766), J. H. Lambert
(1765, 1782), G. J. von Holland (1764), G. F. Castillon (1803) and
J. D. Gergonne (1816/17).* But no school arose.
One who did found a school, and who stands at the beginning of
the continuous development of mathematical logic, is George
Boole, whose first pioneer work, The Mathematical Analysis of
Logic, appeared in 1847. In the same year Augustus de Morgan
published his Formal Logic. Boole's ideas were taken further in
different directions by R. L. Ellis (1863), W. S. Jevons (1864),
R. Grassmann (1872), J. Venn (1880, 1881), Hugh McColl (1877/78),
finally and chiefly by E. Schroder (1877, 1891-95).
Contemporaneous with the last-named are the works of a new
group of mathematical logicians whose chief representatives are
C. S. Peirce (1867, 1870), Gottlob Frege (1879), and G. Peano
(1888). Of these three important thinkers only Peano founded a
considerable school; Peirce and Frege went practically unnoticed.
It was Bertrand Russell (1903) who discovered the thought of
Frege and together with A. N. Whitehead combined it with his
own discoveries in Principia Mathematica (1910-13), in which the
symbolism of Peano was used.
D. Hilbert (1904) and L. E. J. Brouwer (1907, 1908) were active
before the appearance of the Principia. J. Lukasiewicz published
his first work in this field in 1910, St. Lesniewski in 1911. They
were followed by A. Tarski (1921), R. Carnap (1927), A. Heyting
(1929) and K. Godel (1930).
* Figures in parentheses give the year of publication of the main work,
then of the first subsequent important one.
267
MATHEMATICAL LOGIC
These are only a few of the great number of mathematical logi-
cians, which by now is beyond count.
C. FREGE
Among all these logicians, Gottlob Frege holds a unique place.
His Begriffsschrift can only be compared with one other work in
the whole history of logic, the Prior Analytics of Aristotle. The two
cannot quite be put on a level, for Aristotle was the very founder of
logic, while Frege could as a result only develop it. But there is a
great likeness between these two gifted works. The Begriffsschrift,
like the Prior Analytics, contains a long series of quite new insights,
e.g. Frege formulates for the first time the sharp distinction between
variables and constants, the concepts of logical function, of a many-
place function, of the quantifier; he has a notably more accurate
understanding of the Aristotelian theory of an axiomatic system,
distinguishes clearly between laws and rules, and introduces an
equally sharp distinction between language and meta-language,
though without using these terms; he is the author of the theory of
description; without having discovered, indeed, the notion of a
value, he is the first to have elaborated it systematically. And that
is far from being all.
At the same time, and just like Aristotle, he presents nearly all
these new ideas and intuitions in an exemplarily clear and systematic
way. Already in the Begriffsschrift we have a long series of mathe-
matico-logical theorems derived from a few axioms 'without
interruption' (luckenlos), as Frege says, for the first time in history.
Various other mathematical logicians at the same time, or even
earlier, expounded similar ideas and theories, but none of them had
the gift of presenting all at once so many, often quite original,
innovations in so perfect a form.
It is a remarkable fact that this logician of them all had to wait
twenty years before he was at all noticed, and another twenty before
his full strictness of procedure was resumed by Lukasiewicz. In this
last respect, everything published between 1879 and 1921 fell below
the standard of Frege, and it is seldom attained even today. The
fate of Frege's work was in part determined by his symbolism.
It is not true that it is particularly difficult to read, as the reader
can assure himself from the examples given below; but it is certainly
too original, and contrary to the age-old habits of mankind, to be
acceptable.
All that we have said does not mean that Frege is the only great
logician of the period now under consideration. We also have to
recognize as important the basic intuitions of Boole, and many
discoveries of Peirce and Peano, to name only these three. The very
fact that Frege was a contemporary of Peirce and Peano forbids
268
PERIODS
one to treat him as another Aristotle. But of all mathematical
logicians he is undoubtedly the most important.
D. PERIODS
The history of mathematical logic can be divided into four periods.
1. Prehistory: from Leibniz to 1847. In this period the notion of
mathematical logic arose, and many points of detail were formulated,
especially by Leibniz. But there was no school at this time, and
the continuous development had not yet begun. There were, rather,
isolated efforts which went unnoticed.
2. The Boolean period, from Boole's Analysis to Schroder's Vor-
lesungen (vol. I, 1895). During this period there is a continuous
development of the first form of mathematical logic. This form is prin-
cipally distinguished from later ones in that its practitioners did not
make the methods of mathematics their object of study, but con-
tented themselves with simply applying them to logic.
3. The period of Frege, from his Begriffsschrift (1879) to the
Principia Mathematica of Whitehead and Russell (1910-13). Frege,
and contemporaneously Peirbe and Peano, set a new goal, to find
foundations for mathematics. A series of important logical ideas
and methods were developed. The period reaches its peak with the
Principia which both closes the preceding line of development and
is the starting point of a new one, its fruitfulness being due in the
first place to a thorough consideration and solution of the problem
of the antinomies which had been a burning question since the end
of the 19th century and had not previously found a solution in the
new period.
4. The most recent period : since the Principia, and still in progress.
This period can be sub-divided: the years from 1910 to 1930 are
distinguished by the rise of metalogic, finitist in Hilbert, not so
in Lowenheim and Skolem; after about 1930 metalogic is systema-
tized in a formalistic way, and we have Tarski's methodology,
Carnap's syntax, and the semantics of Godel and Tarski in which
logic and metalogic are combined. The 'natural' logics of Gentzen
and Jaskowski (1934) also belong here.
So we can say that the advance of metalogic is distinctive of the
time since 1910, though new logical systems (in the object language)
continue to appear: that of Lewis (1918), the many-valued systems of
Post and Lukasiewicz (1920-21), the intuitionistic logic of Heyting
(1930). Finally the very original systems of combinatorial logic
by Schonfinkel (1924), Curry (1930), Kleene (1934), Rosser (1935)
and Church (1936-41).
This fourth period will be touched on only very lightly, in some
of its problems.
The following table gives an easy view of the temporal sequence
269
MATHEMATICAL LOGIC
of the logicians we have named. But it is to be noticed (1) that
temporal succession does not always reflect actual influence; this
will be discussed more in detail in the various chapters. (2) the subject
developed so fast after 1870 that dates of births and deaths are
little to the purpose ; we have preferred to give those of publication
of the chief logical works.
G. W. v. Leibniz
A. De Morgan 1847
C. S. Peirce 1867-1870
G. Frege 1879
G. Peano 1888
G. Boole 1847
R. C. Ellis 1863
W. S. Jevons 1864
R. Grassmann 1872
H. McColl 1877/78
E. Schroder 1877
D. Hilbert 1904
L. Brouwer 1907/8
B. Russell 1903
Principia 1910-1913
J. Lukasiewicz 1910
St. Lesniewski 1911
A. Tarski 1921
R. Carnap 1927
A. Heyting 1929
K. Godel 1930
E. STATE OF RESEARCH
Mathematical logic is the best known form of logic, since many
of its basic works, especially the Principia, so far from being past
history are still in current use. Then again there have already been
a number of historical studies of the period. Among these are the
work of B. Jourdain (37.01), the historical sections of the works of
C. I. Lewis (37.02) and J. Jorgensen (37.03). The treatise of H. Hermes
and H. Scholz (37.04) is remarkably rich in historical information.
Since 1936 we have had as unique tools of research, a biblio-
graphy of mathematical logic from Leibniz to 1935, and the Journal
of Symbolic Logic containing a current bibliography and excellent
indexes. Both are as good as bibliograj hy can be, under the editor-
ship of A. Church who sees to them with exemplary punctuality and
regularity. Among other contributions to the history of this period
the numerous papers of R. Feys should be mentioned.
270
METHOD
But still we do not know all about the period. L. Couturat's
thorough and serious monograph on Leibniz needs completing on
many points in the light of more recent systematic and historical
research; there are also various other treatises on Leibniz's logic.
Boole, too, has been fairly thoroughly investigated in recent years.
But as yet there is no detailed treatment of Leibniz's successors, no
monograph on Peirce, above all no thorough work on Frege's logic,
without mentioning other less important logicians.
F. METHOD
For the reason stated in the introduction, we have tried to present
the essential range of problems discussed in mathematical logic by
means of texts containing little or no artificial symbols. This has
proved feasible by and large, but not without exception ; in particular,
at least the basic methods have to be explained in terms of the
contemporary symbolism, e.g. of Frege or the Principia. Then again,
we have given the most important theorems in the various fields
in symbolic formulation, in order to facilitate comparison with
similar theorems developed in other periods.
The question of what time-limit to put is very difficult, and the
various periods within the main one dovetail into each other in such
a way as to make the drawing of sharp boundaries impossible. We
have finally decided to close the exposition with the Principia,
touching lightly on a few later developments which are either
closely connected with matters discussed before 1910, or of special
interest on their own account. That section (§ 49) is accordingly in
the nature of an appendix.
The reader will only be able to appreciate the textual fragments that
follow if, first, he is well acquainted with the fundamental concepts
of contemporary mathematical logic (cf. § 5, B); second, he is able to
abstract from the philosophy (ontology, epistemology, psychology
etc.) of the various logicians. For never before have formal logicians
been so divided by mutually opposed philosophies as here. We need
only instance Frege's outspoken Platonism, and Boole's nominalism
and even psychologism. But they have all developed essentially the
same formal logic.
That is not to say that the individual philosophic views have been
entirely without influence on the form of this or that system. But
such influence has been much slighter than an unbiased observer
might at first suppose. That the systems present such different
appearances is due mainly to differences of immediate purpose
(one may compare Boole with Frege, or Peano with Lukasiewicz),
and to differences in the degree of exactness which are more marked
here than in any other period.
271
§38. METHODS OF MATHEMATICAL LOGIC
Two essentially distinct methodological ideas seem to underlie
mathematical logic. On the one hand it is a logic that uses a calculus.
This was developed in connection with mathematics, which at first
was considered as the ideal to which logic should approach. On the
other hand mathematical logic is distinguished by the idea of exact
proof. In this respect it is no hanger-on of mathematics, and this is
not its model; it is rather the aim of logic to investigate the founda-
tions and conduct of mathematics by means of more exact methods
than have been customary among 'pure' mathematicians, and to
offer to mathematics the ideal of strict proof.
In both respects the name 'mathematical logic' is justified,
though for opposite reasons; first, because the new logic is a result of
mathematics, then because it seeks to provide a basis for that
science. But it would be a misunderstanding to conclude that mathe-
matical logicians want to confine themselves to the consideration of
quantities; their aim from the start has rather been to construct a
quite general logic.
In what follows we illustrate both aspects with a series of texts
which resume the development of mathematical logic.
A. LOGICAL CALCULUS
1. Lull
The idea of a mechanical process to facilitate inference is already
present in the combinatorial arguments of the ancient Commentators,
the Arabs and the Scholastics. We have given one example above
(32.34), but of course it was only a matter of determining correct
syllogistic moods. Raymond Lull (1235-1315) is the first to lay claim
to a quite general mechanical procedure. It appears from the work
of this remarkable man that he believed himself to have found a
method which permits one to draw every kind of conclusion by
means of a system of concentric, circular sheets or rings, of various
sizes and mutually adjustable, with letters inscribed on their rims.
Unfortunately Lull does not express the main ideas of this procedure
at all clearly. However, it will be well to give at least a few passages
from his Ars Magna, since his doctrine is not only one of the greatest
curiosities in the history of logic, but also had some influence on
Leibniz.
38.01 The understanding longs and strives for a universal
science of all sciences, with universal principles in which the
principle of the other, more special sciences would be implicit
and contained as is the particular in the universal. . . .
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CALCULUS
38.02 This art is divided into thirteen parts, viz. the
alphabet, the figures, definitions, rules, tables . . . (etc.).
The alphabet of this art is the following:
B signifies goodness, difference, whether, God, justice,
avarice.
C signifies quantity, conformity, what, angel, prudence,
throat.
We spare the reader the further enumeration of this alphabet.
But we print a picture of the 'first figure' and here is part of the
accompanying commentary:
38.03 There are four figures, as appears from this page.
The first figure is signified by A; and it is circular, subdivided
into nine compartments. In the first compartment is B, in
the second C, etc. And it is said to be cruciform, in that the
subject is turned into the predicate and conversely, as when
one says: great goodness, good greatness ; eternal greatness,
great eternity; God the good (Deus bonus), the good God
(bonus Deus), and correspondingly for other (terms). By
means of rotations of this kind the practitioner (artista) can
see what is converted and what is not converted, such as
'God is good' and the like, which can be converted. But God
and angel will not be converted, nor goodness and angel,
nor its goodness and (its) greatness, and so on with the other
terms.
This text is far from clear, and its consequences no clearer; it has,
moreover, little relevance for genuine logic. But the mere idea of such
a mechanical process was a fascinating one for many people in the
16th and 17th centuries.
2. Hobbes
Lull's ideas are to be found expressed in an extreme form three
hundred years later, by Thomas Hobbes (1655). He made no attempt
to carry them out, for like most modern philosophers Hobbes was no
logician.
38.04 By ratiocination I mean computation. Now to
compute, is either to collect the sum of many things that
are added together, or to know what remains when one thing
is taken out of another. Batiocination, therefore, is the same
with addition and substruction (sic); and if any man add
multiplication and division, I will not be against it, seeing
multiplication is nothing but addition of equals one to
273
MATHEMATICAL LOGIC
another, and division nothing but a substraction of equals
one from another, as often as is possible. So that all ratio-
cination is comprehended in these two operations of the mind,
addition and substraction.
This is, to be sure, rather the jeu d' esprit of a dilettante than a
theory of mathematical logic; no inference can be interpreted in this
way, and Hobbes never once tried to do it. The passage shows the
mark of his extreme verbalism, inference being a mere accumulation
of words. However, this text is historically important as having
exercised some influence on Leibniz, and it is also characteristic
of the mathematicism which largely dominated the new form of
logic until Jevons. But perhaps no logician was so badly infected
with it as Hobbes.
3. Leibniz
Leibniz had read Lull (38.05) and cites Hobbes too (38.06). But he
has much more to offer than either ol them. Like Lull, he is con-
cerned with a universal basis for all sciences; like Lull again, his
basic philosophy leads him to think of a purely combinatorial
method. But this is now to take the form of a calculus, such as is
employed in mathematics ; logic is to be thought of as a generalized
mathematics. Leibniz's most characteristic texts on this point are
the following:
38.07 As I was keenly occupied with this study, I happened
unexpectedly upon this remarkable idea, that an alphabet
of human thought could be devised, and that everything
could be discovered (inveniri) and distinguished (dijudicari)
by the combination of the letters of this alphabet and by the
analysis of the resulting words.
38.08 The true method should afford us a filum Ariadnes,
i.e. a sensibly perceptible and concrete means to guide the
mind, like the lines drawn in geometry and the forms of the
operations which are prescribed to learners in arithmetic.
Without this our mind could traverse no path without going
astray.
38.09 To discover and prove truths, the analysis of ideas
is necessary, . . . which corresponds to the analysis of (written)
characters. . . . Hence we can make the analysis of ideas
sensibly perceptible and conduct it as with a mechanical
thread; since the analysis of the characters is something
sensibly perceptible.
274
The "first figure" of Lull's "Ars Magna". Cf. 38.03
CALCULUS
38.10 A characteristic of reason, by means of which truths
would become available to reason by some method of calcula-
tion, as in arithmetic and algebra, so in every other domain,
so long as it submits to the course of deduction.
38.11 Then, in case of a difference of opinion, no discussion
between two philosophers will be any longer necessary, as
(it is not) between two calculators. It will rather be enough
for them to take pen in hand, set themselves to the abacus,
and (if it so pleases, at the invitation of a friend) say to one
another: Calculemus!
38.12 Ordinary languages, though mostly helpful for the
inferences of thought, are yet subject to countless ambiguities
and cannot do the task of a calculus, which is to expose
mistakes in inference owing to the forms and structures of
words, as solecisms and barbarisms. This remarkable advan-
tage is afforded up to date only by the symbols (notae) of
arithmeticians and algebraists, for whom inference consists
only in the use of characters, and a mistake in thought and
in the calculus is identical.
38.13 Hence it seems that algebra and the mathesis uni-
versalis ought not to be confused with one another. If indeed
mathesis was to deal only with quantity, or with equals and
unequals, or with mathematical ratio and proportion, there
would be nothing to prevent algebra (which considers quan-
tity in general) from being considered as their common part.
But mathesis seems to underlie whatever the power of imagina-
tion underlies, insofar as that is accurately conceived, and so
it pertains to it to treat not only of quantity but also of the
arrangement (dispositio) of things. Thus mathesis universalis,
if I am not mistaken, has two parts, the ars combinatoria
concerned with the variety of things and their forms or
qualities in general insofar as they are subject to exact
inference, and the equal and the unequal; then logistic or
algebra, which is about quantity in general.
There are here two different, though connected ideas: that of an
'alphabet of thought' and that of mathesis universalis. According
to the first, one is to assign a symbol to every simple idea and solve
all problems by combinations of these symbols. This is very con-
sonant with Leibniz's philosophy, in particular with his doctrine of
the strictly analytic character of all necessary propositions and of
inference as a combining of elements. This philosophical view,
questionable in itself, was yet fruitful for logic in that it led Leibniz
275
MATHEMATICAL LOGIC
to the notion of an artificial language (38.12) which, by contrast to
ordinary languages, would be free from ambiguities. Therein Leibniz
is the founder of symbolic logic as such, i.e. of the use of artificial
symbols even for logical constants (and not only for variables as in
all earlier forms of logic).
The other idea is that of mathesis universalis (38.10), i.e. of the
application of calculation to all inferences, not only to those that
are mathematical in the narrower sense. Leibniz does not advocate
any mathematicism such as that of Hobbes: malhesis universalis is
sharply distinguished from ordinary algebra (here strangely called
'logistic') and set in contrast to it (38.13). It is only the method that
is to be applied to logic, and this is not any 'addition and subtrac-
tion' as with Hobbes, but simply formal operating with symbols. Of
course the idea of a strictly formalistic logic, of constructing some
meaningless system which is only interpreted later, such as we find in
Boole, is not yet present. The calculus is to be a filum Ariadnes to
assist the mind. The process envisaged is therefore basically the
same as in the earlier logical tradition; formal laws are abstracted
from meaningful sentences. But the principle of a formal process, i.e.
of calculation, is here clearly expressed for the first time, so far as we
know. In this, Leibniz is the founder of mathematical logic.
4. Lambert
Some further development of Leibniz's ideas is to be found in
Lambert (1728-1777):
38.14 Let us see, then, how a more universal idea can be
abstracted from the arithmetical and algebraic calculi. First,
the idea of quantity must be got rid of, as being too special.
You may put in its place qualities, affections, thinys, truths,
ideas and whatever can be discussed, combined, connected,
separated and chanyed into ever new forms; all and each of these
substitutions can be made in accordance with the nature of
things. For each of these operations and changes, with due
differences allowed for, are applied to quantities.
Further, for the ideas of equality, equation, ratio, relation,
proportion, proyression, etc. which occur in arithmetic, more
universal ones are to be substituted. So that in place of
equality it will be convenient to introduce identity, in place of
equation identification, if this word is taken in its active sense,
in place of proportion analoyy. And if the words relation,
proyression be retained, their meaning is to be extended, as
ordinary usage suggests, so that they can be thought of as
relations or progressions between the thinys, qualities,
276
CALCULUS
affections, ideas or truths to which the calculus is to be suited.
And this is to be chiefly noted, that those relations contribute
no little to determining the form of the calculus, and that all
those operations which the object of the calculus admits rest chiefly
on them.
5. Gergonne
Gergonne (1816/17) comes much closer to the idea of formalism:
38.15 It is constantly being said that reasoning must only
be about objects of which one has a perfectly clear idea, yet
often nothing is more false. One reasons, in practice, with
words, just as one calculates with letters in algebra; and in
the same way that an algebraic calculation can be carried out
exactly without one having the slightest idea about the sig-
nification of the symbols on which one is operating, in the
same way it is possible to follow a course of reasoning without
any knowledge of the signification of the terms in which it is
expressed, or without adverting to it if one knows it. . . . It is
evident, for example, that one does not have to know the
nature of the terms of a proposition in order to deduce its
converse or subaltern when it admits of such. Doubtless one
cannot dispense with a good knowledge of notions which are to
be the immediate matter of judgment; but that is quite
unnecessary for concluding to a judgment from a number of
others already known to be correct.
This text is not altogether clear; Gergonne seems to equate the
(Aristotelian) use of variables with formalism. But we can see the
idea of formalism becoming clearer.
Gergonne also gave expression to an idea which is not without
relevance to the symbolism of mathematical logic:
38.16 There is no known language in which a proposition
exactly and exclusively expresses in which of our five relations
both its component terms stand; such a language would have
five kinds of proposition and its dialectic would be quite
different from that of our languages.
He is referring there to five relationships between the extensions
of two terms (or classes) which will be spoken of later (40.12). *
* We learned of this passage from a work of J. A. Faris.
277
MATHEMATICAL LOGIC
6. Boole
We can find a clear idea of formalism, developed in an exemplary
way, in the introduction to George Boole's epoch-making The
Mathematical Analysis of Logic (1847), in this superior to much later
works, e.g. the Principia.
38.17 They who are acquainted with the present state of
the theory of Symbolical Algebra, are aware, that the validity
of the processes of analysis does not depend upon the inter-
pretation of the symbols which are employed, but solely upon
the laws of their combination. Every system of interpretation
which does not affect the truth of the relations supposed, is
equally admissible, and it is thus that the same process may,
under one scheme of interpretation, represent the solution of a
question on the properties of numbers, under another, that
of a geometrical problem, and under a third, that of a problem
of dynamics or optics. This principle is indeed of fundamental
importance; and it may with safety be affirmed, that the
recent advances of pure analysis have been much assisted by
the influence which it has exerted in directing the current of
investigation.
But the full recognition of the consequences of this impor-
tant doctrine has been, in some measure, retarded by acciden-
tal circumstances. It has happened in every known form of
analysis, that the elements to be determined have been con-
ceived as measurable by comparison with some fixed standard.
The predominant idea has been that of magnitude, or more
strictly, of numerical ratio. The expression of magnitude, or
of operations upon magnitude, has been the express object
for which the symbols of Analysis have been invented, and
for which their laws have been investigated. Thus the abstrac-
tions of the modern Analysis, not less than the ostensive
diagrams of the ancient Geometry, have encouraged the
notion, that Mathematics are essentially, as well as actually,
the Science of Magnitude.
The consideration of that view which has already been
stated, as embodying the true principle of the Algebra of
Symbols, would, however, lead us to infer that this conclusion
is by no means necessary. If every existing interpretation is
shewn to involve the idea of magnitude, it is only by induction
that we can assert that no other interpretation is possible.
And it may be doubted wither our experience is sufficient to
278
CALCULUS
render such an induction legitimate. The history of pure
Analysis is, it may be said, too recent to permit us to set
limits to the extent of its applications. Should we grant to the
inference a high degree of probability, we might still, and with
reason, maintain the sufficiency of the definition to which the
principle already stated would lead us. We might justly
assign it as the definitive character of a true Calculus, that it is
a method resting upon the employment of Symbols, whose
laws of combination are known and general, and whose
results admit of a consistent interpretation. That to the
existing forms of Analysis a quantitative interpretation is
assigned, is the result of the circumstances by which those
forms were determined, and is not to be construed into a
universal condition of Analysis. It is upon the foundation of
this general principle, that I purpose to establish the Calculus of
Logic, and that I claim for it a place among the acknowledged
forms of Mathematical Analysis, regardless that in its object
and in its instruments it must at present stand alone.
From that Boole draws the explicit conclusion:
38.18 On the principle of a true classification, we ought no
longer to associate Logic and Metaphysics, but Logic and
Mathematics. . . . The mental discipline which is afforded by
the study of Logic, as an exact science, is in species, the same
as that afforded by the study of Analysis.
Leibniz and Lambert had already wanted to apply calculation
to logic, and had used the idea of non-quantitative calculation. The
epoch-making feature of Boole's text is the exemplarily clear account
of the essence of calculation, viz. formalism, a process of which the
'validity does not depend upon the interpretation of the symbols
which are employed, but solely upon the laws of their combination'.
Boole is also aware of the possibility of interpreting the same formal
system in different ways. This suggests that he did not think of logic
as an abstraction from actual processes, as all previous logicians had
done, but as a formal construction for which an interpretation is
sought only subsequently. That is quite new, and in contrast with
the whole tradition, Leibniz included.
7 '. Peirce
Finally we submit a text from Peirce's review of Schroder's logic
(1896), which contains one of the best statements of the advantage
to be looked for in a logical calculus.
38.19 It is a remarkable historical fact that there is a
branch of science in which there has never been a prolonged
279
MATHEMATICAL LOGIC
dispute concerning the proper objects of that science. It is
mathematics. Mistakes in mathematics occur not infrequently,
and not being detected give rise to false doctrine, which may
continue a long time. Thus, a mistake in the evaluation of a
definite integral by Laplace, in his Mecanique celeste, led to an
erroneous doctrine about the motion of the moon which
remained undetected for nearly half a century. But after the
question had once been raised, all dispute was brought to a
close within a year. . . .
38.20 Hence, we homely thinkers believe that, considering
the immense amount of disputation there has always been
concerning the doctrines of logic, and especially concerning
those which would otherwise be applicable to settle disputes
concerning the accuracy of reasonings in metaphysics, the
safest way is to appeal for our logical principles to the science
of mathematics, where error can only long go unexploded on
condition of its not being suspected. . . .
38.21 Exact logic will be that doctrine of the conditions of
establishment of stable belief which rests upon perfectly
undoubted observations and upon mathematical, that is,
upon diagrammatical, or iconic, thought. We, who are sectaries
of 'exact' logic, and of 'exact' philosophy, in general, main-
tain that those who follow such methods will, so far as they
follow them, escape all error except such as will be speedily
corrected after it is once suspected.
B. THEORY OF PROOF
1. Bolzano
A noteworthy precursor of modern proof-theory is Bernard
Bolzano*.
38.22 If we now state that M, N, 0, . . . are deducible from
A, B, C, . . . and this in respect of the notions i, /, . . .: we are
basically saying, according to what has been said in § 155,
the following: 'All ideal contents which in the place of i, /, . . .
in the propositions A, B, C, . . . M, N, 0, . . . simultaneously
verify the propositions A, B, C, . . . has the property of also
simultaneously verifying the propositions M, TV, 0. . . .' The
* Professor Hans Hermes drew our attention to this passage.
The rows of dots after the groups of letters are here part of the text.
280
PROOF
most usual way of giving expression to such propositions is of
course: '// A, B, C, . . . are true: then also M, TV, 0, . . . are
true.' But we often also say: 'M, TV, 0, . . . follow, or are Reduc-
ible, or can be inferred from A, B, C, . . . etc' In respect
of the notions i, /',... which we consider as the variables in
these propositions, the same remark is applicable as in No. 1.
But since according to § 155 No. 20 it is not at all the < •;)-«■
with the relation of deducibility, as (it is) with the relations of
mere compatibility, that a given content of propositions
A, B, C, .... on the one hand, and M, TV, 0, ... on the other,
can come into this relationship merely because we determine
arbitrarily which notions therein are to count as variables:
it is thus a rather startling statement when we say that certain
propositions M, N, 0, . . . can be brought into a relationship of
deducibility with other propositions A, B, C, ... by merely
taking the notions pertaining to them as variable. But in such
a judgment we only say that there are certain parts of the
propositions A, B, C, . . . M, TV, 0, . . . which can be considered
as variable, with the result that every ideal content which in
the place of i, /, . . . makes all of A, B, C, . . . true, also makes
all of M, TV, 0, . . . true. And thus we can easily see from
§ 137 how such a proposition must be expressed to bring out
its logically constant parts. 'The notion of some parts of
A, B, C, . . . M, TV, 0, ... so constituted that every arbitrary
ideal content which in their place verifies A, B, C, . . . always
also verifies M, TV, 0, . . . has objectivity'. In ordinary speech
propositions of this kind are expressed just like the preceding
ones. It is only from other circumstances, e.g. from the
context, that one can guess whether the speaker has in his
mind determinate notions in respect to which the retation of
deducibility is to be present, or whether he only inlends to
intimate that there are such notions. Thus, e.g., it is easy
enough to gather from the following proposition: 'if Caius is a
man, and all men are mortal, then Caius, too, is mortal', that
it is here intended to state the deducibility of the proposition :
Caius is mortal, from the two propositions : Caius is a man, and,
all men are mortal, in respect of the three notions; Caius, man
and mortal. This next utterance on the other hand: 'If in all
men there stirs an irresistible desire for permanence ; if, too, the
most virtuous must feel unhappy at the thought that he is
one day to cease; then we are not wrong to expect of God's
infinite goodness that he will not annihilate us in death' -
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MATHEMATICAL LOGIC
this would be subject to the reproach of extreme obscurity,
since its sense is not that the said propositions stand in a
relationship of deducibility when some of their notions (which
still have to be ascertained) have been taken as variable.
By such an utterance it is only intended to state that notions
are present such as to warrant inference from the truth of the
antecedent to the truth of the consequent; but it does not as
yet tell one which these notions properly are.
2. Frege
While that text of Bolzano's contains important ideas about the
concept of deduction or deducibility, the modern development of this
second aspect of mathematical logic begins with Frege. We take the
essential texts from the Grundgesetzen der Arilhmeiik (1893); but it
can easily be shown that most of what is said in them was already
known to this great logician by 1879.
38.23 In my Grundlagen der Arithmetik I have tried to
make it plausible that arithmetic is a branch of logic and does
not need to take its grounds of proof either from experience or
intuition. This will now be confirmed in the present book, by
the fact that the simplest laws of numbers can be deduced by
logical means alone. But at the same time this shows that
considerably higher demands must be made on the process of
proof than is usual in arithmetic. A region of some ways of
inference and deduction must be previously delimited, and no
step may be made which is not in accordance with one of
these. In the passage, therefore, to a new judgment, one must
not be satisfied with the fact that it is evidently correct, as
mathematicians nearly always have been up to now, but one
must analyze it into its simple logical steps, which are often by
no means a few. No presupposition may remain unremarked ;
every axiom which is needed must be discovered. It is just the
tacit presuppositions, that are made without clear conscious-
ness, which obscure understanding of the epistemological cha-
racter of a law.
38.24 The ideal of a strict scientific method in mathematics,
such as I have here tried to realize, and which could well be
called after Euclid, I might describe thus. It cannot indeed
be required that everything should be proved, since that is
impossible; but one can see to it that all propositions which
are used without being proved, are expressly stated as such,
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PROOF
so that it is clearly known on what the whole structure rests.
The effort, then, must be to reduce the number of these
primitive laws as far as possible, by proving everything which
can be proved. Further, and here I go beyond Euclid, I
require that all methods of inference and deduction which are
to be applied, shall be previously presented. Otherwise it is
impossible to ensure with certainty that the first requirement
is fulfilled. I think that I have now attained this ideal in
essentials. Stricter requirements could only be made in a few
points. In order to secure greater mobility, and not to fall into
excessive prolixity, I have allowed myself to make tacit use of
the commutability of antecedents, and of the identification of
like antecedents, and have not reduced the ways of inference
and deduction to the smallest number. Those who know my
small book Begriffsschrift will be able to gather from it how
the strictest requirements could be forthcoming here as well,
but also that this would bring with it a notable increase in
size.
Frege is correct here in claiming Euclid as his predecessor, insofar
as Euclid was the first to carry out the idea of an axiomatic system in
mathematics. But it would have been much better to refer to Aristotle
(14.02, 14.05), for what Frege offers is an important sharpening of the
Aristotelian concept of an axiomatic system. His first requirement is
that all presuppositions should be formulated expressly and without
gaps. Then he makes an explicit distinction between the laws and the
methods of inference and deduction, i.e. the rules of inference. This
is not altogether new (cf. 22.12-22.15, 30.11. § 31, C), but is stated
with greater clarity than ever before. Finally. Frege can be con-
trasted with Leibniz, Boole and other earlier writers in his laying
down of a quite new requirement: 'considerably higher demands
must be made on the notion of proof than is usual in arithmetic'.
With that, mathematical logic enters on its second phase.
38.23 and 38.24, along with the citations from Boole, are texts
of far-reaching influence on the concept of mathematical logic. In
this connection two further quotations, dating from 1896, may be
added.
38.25 Words such as 'therefore', 'consequently', 'since'
suggest indeed that inference has been made, but say nothing
of the principle in accordance with which it has been made, and
could also be used without misuse of words where there is no
logically justified inference. In an inquiry which I here have
in view, the question is not only whether one is convinced of
the truth of the conclusion, with which one is usually satisfied
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MATHEMATICAL LOGIC
in mathematics ; but one must also bring to consciousness the
reason for this conviction and the primitive laws on which it
rests. Fixed lines on which the deductions must move are
necessary for this, and such are not provided in ordinary
language.
38.26 Inference is conducted in my symbolic system
(Begriffsschrift) according to a kind of calculation. I do not
mean this in the narrow sense, as though an algorithm was in
control, the same as or similar to that of ordinary addition
and multiplication, but in the sense that the whole is algorith-
mic, with a complex of rules which so regulate the passage
from one proposition or from two such to another, that nothing
takes place but what is in accordance with these rules. My
aim, therefore, is directed to continuous strictness of proof and
utmost logical accuracy, along with perspicuity and brevity.
Frege's program of thorough proof was later carried out in mathe-
matics by Hilbert with a view to pure formality. The texts can be
referred to in 0. Becker (38.27). It was Lukasiewicz who applied
it to logical systems with complete strictness. We shall give an
example in the chapter on propositional logic (43.45).
C. METALOGIC
The idea of a metalogic was an inevitable result of the combination
of Boole's formalism and Frege's theory of proof. For once formulae
had been distinguished from rules, and the former treated with strict
formalism, 'after the fashion of an algorithm' as Frege says, then the
rules had to be interpreted as meaningful and having content. At
once the rules are seen as belonging to a different level to the for-
mulae. The notion of this second level appears first in connection
with mathematics as that of metamathematics in Hilbert. We cite his
lecture Die logischen Grundlagen der Malhematik (1923):
38.28 The basic idea of my theory of proof is this:
Everything that goes to make up mathematics in the
accepted sense is strictly formalized, so that mathematics
proper, or mathematics in the narrower sense, becomes a
stock of formulae. . . .
Beyond mathematics proper, formalized in this way, there
is, so to speak, a new mathematics, a metamathematics, which
is needed to establish the other securely. In it, by contrast to
the purely formal ways of inference in mathematics proper,
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PROOF
inference which has regard to the subject matter is applied,
though merely to establish the freedom from contradiction of
the axioms. In this metamathematics we operate with the
proofs of mathematics proper, these last themselves forming
the object of the inference that regards the matter. In this
way the development of the total science of mathematics is
achieved by a continual exchange which is of two kinds: the
gaining of new provable formulae from the axioms by means of
formal inference, and on the other hand the addition of new
axioms along with the proof of freedom from contradiction by
means of inference having regard to the matter.
The axioms and provable propositions, i.e. formulae, which
arise in this process of exchange, are representations of the
thoughts which constitute the usual processes of mathematics
as understood up to now, but they are not themselves truths
in an absolute sense. It is the insights which are afforded by my
theory of proof in regard to provability and freedom from
contradiction which are rather to be viewed as the absolute
truths.
This important text goes beyond the bounds of this chapter in
that it touches not only on proof-theory but also on the concept of
logic and its relations to mathematics, since Hilbert here limits
meaningful inference to the proof of freedom from contradiction,
in accordance with his special philosophy of mathematics. The
important point for our purpose is chiefly the sharp distinction
between the formalized, and so in Boole's sense meaningless,
calculus on the one hand, and the meaningful rules of inference on
the other. This idea, too, was first expressed by Frege, when he requir-
ed enumeration of all 'ways of inference and deduction' as distinct
from axioms (cf. 38.24). But Frege did not think of the axioms and
theorems as meaningless, however formally he considered them. Here
on the contrary it is a case of inscriptions considered purely mate-
rially.
A new stage in the understanding of formalization has thus been
reached. The doctrine in Hilbert is, of course, limited to mathematics -
he speaks of metamathematics. But soon this idea was to be extended
to logic, and this came about in the Warsaw School. The expression
'metalogic' first occurs in a paper by Lukasiewicz and Tarski of
1930 (38.29).
Parallel to the work of the Warsaw School is that which R. Carnap
was carrying on in Vienna at the same time.
We cite now a text of Tarski's, the founder of systematic meta-
logic. He forms the starting-point for the most recent developments
which will not be pursued here.
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MATHEMATICAL LOGIC
Tarski wrote in 1930:
38.30 Our object in this communication is to define the
meaning, and to establish the elementary properties, of some
important concepts belonging to the methodology of the deductive
sciences, which, following Hilbert, it is customary to call
metamathematics .
Formalized deductive disciplines form the field of research of
metamathematics roughly in the same sense in which spatial
entities form the field of research in geometry. These discip-
lines are regarded, from the standpoint of metamathematics,
as sets of sentences. Those sentences which (following a sug-
gestion of S. Lesniewski) are also called meaningful sentences,
are themselves regarded as certain inscriptions of a well-
defined form. The set of all sentences is here denoted by the
symbol iSi. From the sentences of any set X certain other
sentences can be obtained by means of certain operations
called rules of inference. These sentences are called the conse-
quences of the set X. The set of all consequences is denoted by
the symbol lCn {X)\
An exact definition of the two concepts, of sentence and of
consequence, can be given only in those branches of meta-
mathematics in which the field of investigation is a concrete
formalized discipline. On account of the generality of the
present considerations, however, these concepts will here be
regarded as primitive and will be characterized by means of
a series of axioms.
§39. THE CONCEPT OF LOGIC
As has been seen above (§ 38) Boole (38.17), Peirce (38.19) and
with them the other mathematical logicians of the 19th century
considered logic to be a branch of mathematics, this last being
described not with reference to its object but its method, the
application of a calculus. However, at the end of the 19th century
there arose considerable disagreement about the relationship of
logic to mathematics, a disagreement which at the same time
concerned the answer to the question whether logic can be deve-
loped purely formally as a system of symbols, or whether it necessa-
rily involves an interpretation of the symbols. So there were two
problems, but both concerned with the concept of logic. Three main
positions took shape: the logistic, the formalistic (not in the sense
in which 'formalism' is used in the last and in subsequent sections)
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CONCEPT OF LOGIC
and the intuitionistic. We shall illustrate their main features with
some texts.
A. THE LOGISTIC POSITION
On the logistic position there is no essential distinction between
logic and mathematics, inasmuch as mathematics can be developed
out of logic; more exactly, inasmuch as all mathematical terms
can be defined by logical ones, and all mathematical theorems
can be deduced from true logical axioms. Frege is the originator of
this line of thought, which attained its fullest development in the
Principia Malhematica of Whitehead and Russell, written precisely
to provide a thorough proof of the logistic thesis.
1. Frege: semantics
Frege's theory of logic is closely connected with his semantics
(a word which we always use here in the sense of Morris (5.01), not
in Tarski's technical sense). On this point we shall here recall
briefly only that logic for Frege was not a game with symbols but a
science of objective thoughts (Gedanken), i.e. of ideal propositions
(and so of lecta in the sense of 19.04ff.). The premisses must be true,
formalism is only a means. To begin with, we give a text about the
first point:
39.01 By the word 'sentence' (Satz) I mean a sign which is
normally composite, regardless of whether the parts are
spoken words or written signs. This sign must naturally have
a sense (Sinn). I shall here only consider sentences in which we
assert or state something. We can translate a sentence into
another language. In the other language the sentence is
different from the original one, since it consists of different
components (words) differently compounded; but if the
translation is correct, it expresses the same sense. And the
sense is properly just that which matters to us. The sentence
has a value for us through the sense which we apprehend in
it, and which we recognize as the same in the translation too.
This sense I call 'thought' (Gedanke). What wre prove is not
the sentence but the thought. And it makes no difference what
language we use for that purpose. In mathematics people
speak indeed of a proof of a Lehrsaiz when they understand
by the word Satz what I call 'thought' - or perhaps they do not
sufficiently distinguish between the verbal or symbolic
expression and the thought expressed. But for clarity it is
better to make this distinction. The thought is not perceptible
to the senses, but we give it an audible or visible represen-
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MATHEMATICAL LOGIC
tative in the sentence. Hence I say 'theorem' rather than
'sentence', 'axiom' rather than 'primitive sentence', and by
theorems and axioms I understand true thoughts. This further
implies that thoughts are not something subjective, the
product of our mental activity; for the thought, such as we
have in the theorem of Pythagoras, is the same for everyone,
and its truth is quite independent of whether it is or is not
thought by this or that man. Thinking is to be viewed not as
the production of thought but as its understanding.
Here, in another terminology, we have exactly the Stoic doctrine
that logic deals with lecta, and the third scholastic view (28.17)
according to which propositions stand for ideal structures.
On the question of the truth of premisses, Frege says :
39.02 Nothing at all can be deduced from false premisses.
A mere thought which is not accepted as true, cannot be a
premiss. Only when I have accepted a thought as true can it
be a premiss for me; mere hypotheses cannot be used as
premisses. Of course, I can ask what consequences follow from
the supposition that A is true without having accepted the
truth of A ; but the result then involves the condition : if
A is true. But that is only to say that A is not a premiss, since
a true premiss does not occur in the judgment inferred.
Frege thus holds a kind of absolutest doctrine closely approximat-
ing to the Aristotelian theory of a7c68ei£i<; (14.02) but apparently
still more radical.
We append now a characteristic text about the use of quotation-
marks, in which Frege's high degree of exactness finds expression - a
degree that has been too seldom attained since.
39.03 People may perhaps wonder about the frequent use
of quotation-marks ; I use them to distinguish the cases where
I am speaking of the symbol itself, from those where I am
speaking of what it stands for. This may seem very pedantic,
but I consider it necessary. It is extraordinary how an inexact
manner of speaking and writing, which was originally perhaps
used only for convenience, can in the end lead thought
astray after one has ceased to notice it. Thus it has come about
that numerals are taken for numbers, names for what they
name, what is merely auxiliary for the proper object of
arithmetic. Such experiences teach us how necessary it is to
demand exactness in ways of talking and writing.
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CONCEPT OF LOGIC
2. Frege: Logic and Mathematics
39.04 Under the name 'formal theory' I shall here consider
two modes of conception, of which I subscribe to the first and
endeavour to refute the second. The first says that all arith-
metical propositions can, and hence should, be deduced from
definitions alone by purely logical means. . . . Out of all the
reasons which support this view I shall here adduce only one,
which is based on the comprehensive applicability of arith-
metical doctrines. One can in fact number pretty well every-
thing that can be an object of thought: the ideal as well as the
real, concepts and things, the temporal and the spatial, events
and bodies, methods as well as propositions; numbers them-
selves can be in turn numbered. Nothing is actually required
beyond a certain definiteness of delimitation, a certain logical
completeness. From this there may be gathered no less than
that the primitive propositions on which arithmetic is based,
are not to be drawn from a narrow domain to the special
character of which they give expression, as the axioms of
geometry express the special character of the spatial domain;
rather must those primitive propositions extend to everything
thinkable, and a proposition of this most universal kind is
rightly to be ascribed to logic.
From this logical or formal character of arithmetic I draw
some conclusions.
First: no sharp boundary between logic and arithmetic
is to be drawn; considered from a scientific point of view
both constitute a single science. If the most universal primi-
tive propositions and perhaps their immediate consequences
are attributed to logic, and the further development to
arithmetic, it is like wanting to detach a special science of
axioms from geometry. Yet the partitioning of the whole
domain of knowledge among the sciences is determined not
only by theoretical but also by practical considerations, so
that I do not wish to say anything against a certain practical
separation. But it must not become a breach as is now the
case to the detriment of both. If this formal theory is correct,
logic cannot be so fruitless as it may appear to a superficial
consideration - of which logicians are not guiltless. And there
is no need for that attitude of reserve on the part of many
mathematicians towards any philosophic justification of
whatever is real, at least insofar as it extends to logic. This
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MATHEMATICAL LOGIC
science is capable of no less exactness than mathematics
itself. On the other hand logicians may be reminded that they
cannot learn to know their own science thoroughly if they
do not trouble themselves about arithmetic.
39.05 My second conclusion is that there is no special
arithmetical kind of inference such that it cannot be reduced
to the common inference of logic.
39.06 My third conclusion concerns definitions, as my
second concerned kinds of inference. In every definition
something has to presupposed as known, by means of which
one explains what is to be understood by a name or symbol.
An angle cannot be well defined without presupposing know-
ledge of a straight line. Now that on which a definition is
based may itself be defined; but in the last resort one must
always come to something indefinable, which has to be
recognized as simple and incapable of further resolution. And
the properties which belong to these foundation stones of
science, contain its whole content in embryo. In geometry
these properties are expressed in the axioms, to the extent
that these are independent of one another. Now it is clear
that the boundaries of a science are determined by the
nature of its foundation stones. If, as in geometry, we are
originally concerned with spatial structures, the science, too,
will be limited to what is spatial. Since then arithmetic is to
be independent of all particular properties of things, that
must hold for its foundations : they must be of a purely logical
kind. The conclusion follows that everything arithmetical
is to be reduced by definitions to what is logical.
3. Russell
Frege's postulates were first taken up by Giuseppe Peano - though
without direct dependence on Frege - then by Bertrand Russell.
The latter extended the logistic thesis to geometry and mathematical
disciplines in general.
39.07 The general doctrine that all mathematics is deduc-
tion by logical principles from logical principles was strongly
advocated by Leibniz. . . . But owing partly to a faulty logic,
partly to belief in the logical necessity of Euclidean Geometry,
he was led into hopeless errors. . . . The actual propositions
of Euclid, for example, do not follow from the principles of
logic alone; .... But since the growth of non-Euclidean
Geometry, it has appeared that pure mathematics has no
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CONCEPT OF LOGIC
concern with the question whether the axioms and propositions
of Euclid hold of actual space or not: this is a question for
applied mathematics, to be decided, so far as any decision is
possible, by experiment and observation. What pure mathe-
matics asserts is merely that the Euclidean propositions follow
from the Euclidean axioms - i.e. it asserts an implication:
any space which has such and such properties has also such
and such other properties. Thus, as dealt with in pure mathe-
matics, the Euclidean and non-Euclidean Geometries are
equally true: in each nothing is affirmed except implica-
tions. . . .
39.08 Thus pure mathematics must contain no indefinables
except logical constants, and consequently no premisses, or
indemonstrable propositions, but such as are concerned
exclusively with logical constants and with variables. It is
precisely this that distinguishes pure from applied mathe-
matics.
How and to what extent this program was carried out, cannot
here be pursued. Reference may be made to Becker (39.09). In
conclusiou we should like only to illustrate Frege's definition of
number by means of purely logical concepts, especially with a view
to comparing it with a similar discovery in the Indian logic of the
17th century (54.17).
4. Frege: number
39.10 To illuminate matters it will be good to consider
number in connection with a judgment where its primitive
manner of application occurs. If when I see the same outward
appearances I can say with the same truth: 'this is a group of
trees' and 'these are five trees' or 'here are four companies'
and 'here are 500 men', no difference is made to the individual
or to the whole, the aggregate, but to my naming. But this
is only the sign of the substitution of one concept by another.
This suggests an answer to the first question of the previous
paragraph, that number involves a statement about a concept.
This is perhaps most evident for the number 0. When I say:
'Venus has 0 moons', there is no moon or aggregate of moons
there about which anything can be said; but to the concept
'moon of Venus' there is attributed a property, viz. that of
comprising nothing under it. When I say: 'the emperor's
carriage is drawn by four horses', I apply the number four to
the concept 'horse which draws the emperor's carriage'. . . .
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MATHEMATICAL LOGIC
39.11 Among the properties which are predicated of a
concept I do not, of course, understand the notes which make
up the concept. These are properties of the things which fall
under the concept, not of the concept. Thus 'right-angled'
is not a property of the concept 'right-angled triangle'; but
the proposition that there is no right-angled, rectilineal,
equilateral triangle, states a property of the concept 'right-
angled, rectilineal, equilateral triangle', attributing to it the
number 0.
39.12 In this respect existence is like number. The affir-
mation of existence is nothing else than the denial of the
number 0. Since existence is a property of the concept, the
ontological proof of the existence of God fails of its purpose
It would also be false to deny that existence and unicity
can ever be notes of concepts. They are only not notes of that
concept to which the manner of speech might lead one to
ascribe them. E.g. when all concepts belonging only to one
object are collected under one concept, uniqueness is a note of
this concept. The concept 'moon of the earth', for instance,
would fall under it, but not the so-called heavenly body.
Thus a concept can be allowed to fall under a higher one,
under a concept, so to speak, of second order. But this rela-
tionship is not to be confused with that of subordination.
Frege's definition of number was later interpreted by Russell
extensionally, when he took numbers as classes of classes (39.13).
B. FORMALISM
The formalists, too, see no essential difference between logical
and mathematical formulae, but they understand both forma-
listically and think of the single system composed of them as a
system of symbols. Evidence and truth of the axioms have no part
to play: but freedom from contradiction is everything. The founder
of formalism is David Hilbert, the essentials of whose thought on
logic is contained in the text given earlier (38.28). Here we add only
a brief passage from a letter to Frege in 1899 or 1900:
39.14 You write: 'From the truth of the axioms it follows
that they do not contradict one another'. I was very interested
to read this particular sentence of yours, because for my part,
ever since I have been thinking, writing and lecturing about
such matters, I have been accustomed to say just the reverse:
if the arbitrarily posited axioms are not in mutual (sic) con-
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CONCEPT OF LOGIC
tradiction with the totality of their consequences, then they
are true - the things defined by the axioms exist. That for
me is the criterion of truth and existence.
For the rest, it is not easy to find texts to illustrate Hilbert's
thought before 1930; for that and the later development Becker
may again be consulted (39.15).
It should be noted that formalism has been very important for the
concept of logic, quite apart from its value as a theory. Logic having
been previously viewed as a calculus, it is henceforth ever more and
more transposed onto the level of metalogic. After Hilbert, it is not
the formulae themselves but the rules of operation by which they
are formed and derived that are more and more made the object of
logical investigation.
C. INTUITIONISM
By contrast to the logisticians and formalists, the intuition ists
make a sharp distinction between logic and mathematics. Mathe-
matics is not, for them, a set of formulae, but primarily a mental
activity the results of which are subsequently communicable by
means of language. In language, as used by mathematicians,
certain regularities are observed, and this leads to the development
of a logic. Thus logic is not presupposed by, but abstracted from
mathematics. Once that has been done, it can then be formalized,
but this is a matter of secondary importance. *
Intuitionism has a fairly long history in mathematics: L. Kron-
ecker and H. Poincare are precursors; H. Weyl is reckoned a
'semi-intuitionist'. But L. E. J. Brouwer ranks as the founder of
the school, and intuitionistic logic was first properly formulated
(and formalized) by A. Heyting in 1930.
From the standpoint of formal logic it is to be noted that the
intuitionists, as they themselves say, admit the principle of tertium
exclusion only under certain limitations. In this respect their
doctrine belongs to those 'heterodox' logics of which we shall speak
in § 49.
We give one text from Heyting and one from Brouwer :
39.16 Intuitionistic mathematics is an activity of thought.
and every language - even the formalistic - is for it only a
means of communication. It is impossible in principle to
establish a system of formulae that would have the same
value as intuitionistic mathematics, since it is impossible to
* Special thanks are due to Prof. E. W. Beth for much information in this
connection, as generally for his help with the composition of this fifth part.
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MATHEMATICAL LOGIC
reduce the possibilities of thought to a finite number of
rules that thought can previously lay down. The endeavour
to reproduce the most important parts of mathematics in a
language of formulae is justified exclusively by the great
conciseness and defmiteness of this last as compared with
customary languages, properties which fit it to facilitate
penetration of the intuitionistic concepts and their applica-
tion in research.
For constructing mathematics the statement of universally
valid logical laws is not necessary. These laws are found as it
were anew in every individual case to be valid for the mathe-
matical system under consideration. But linguistic com-
munication moulded according to the needs of everyday life
proceeds according to the form of logical laws which it
presupposes as given. A language which imitated the process
of intuitionistic mathematics step by step would so diverge in
all its parts from the usual pattern that it would have to
surrender again all the useful properties mentioned above.
These considerations have led me to begin the formalization
of intuitionistic mathematics once again with a propositional
calculus.
The formulae of the formalistic systems come into being by
the application of a finite number of rules of operation to a
finite number of axioms. Besides 'constant' symbols they also
contain variables. The relationship between this system and
mathematics is this, that on a determinate interpretation
of the constants and under certain restrictions on substitu-
tion for variables every formula expresses a correct mathe-
matical proposition. (E.g. in the propositional calculus the
variables must be replaced only by senseful mathematical
sentences.) If the system is so constructed as to fulfil the
last-mentioned requirement, its freedom from contradiction
is thereby guaranteed, in the sense that it cannot contain
any formula which would express a contradictory proposition
on that interpretation.
The formalistic system can also be considered mathemati-
cally for its own sake, without reference to any interpretation.
Freedom from contradiction then takes on a new meaning
inasmuch as contradiction is defined as a definite formula;
for us this method of treatment is less to the fore than the
other. But here questions come in about the independence and
completeness of the axiom-system.
294
CONCEPT OF LOGIC
39.17 The differences about the Tightness of the new
formalistic foundations and the new intuitionistic construc-
tion of mathematics will be removed, and the choice between
the two methods of operation reduced to a matter of taste,
as soon as the following intuitions (Einsichten) have been
generally grasped. They primarily concern formalism, but
were first formulated in intuitionist literature. This grasp
is only a matter of time, since they are results purely of
reflection, containing nothing disputable, and necessarily
acknowledged by everyone who has once understood them.
Of the four intuitions this understanding and acknowledge-
ment has so far been attained for two in the formalistic
literature. Once the same state of affairs has been reached for
the other two, an end will have been put to disputes about
foundations in mathematics.
FIRST INTUITION. The distinction between the formalistic
endeavours to construct the 'mathematical slock of formulae'
(formalistic idea of mathematics) and an intuitive (meaningful)
theory of the laws of this construction, as also the understanding
that for the last theory the intuitionistic mathematics of the set
of natural numbers is indispensable.
SECOND INTUITION. The rejection of the thoughtless
application of the logical theorem of tertium exclusum, as also
the awareness first, that investigation of the credentials and
domain of validity of the said theorem constitutes an essential
object of mathematical foundational research; second, that this
domain of validity in intuitive (meaningful) mathematics com-
prises only finite systems.
THIRD INTUITION. The identification of the theorem of
tertium exclusum with the principle of the solubility of every
mathematical problem.
FOURTH INTUITION. The awareness that the (meaningful)
justification of formalistic mathematics through proof of its
freedom from contradiction involves a vicious circle, since this
justification depends on the (meaningful) correctness of the
proposition that the correctness of a proposition follows from
the freedom from contradiction of this proposition, i.e. from the
(meaningful) correctness of the theorem of tertium exclusum.
295
II. THE FIRST PERIOD
§40. THE BOOLEAN CALCULUS
The system of mathematical logic inaugurated by Boole in 1847
holds a special place in history in that it admits of two interpreta- /
tions, in class-logic and propositional logic. In this section we shall ^
consider the abstract calculus itself and its classical interpretation,
reserving the propositional interpretation to the following section.
The growth of Boole's calculus can be summarized as follows:
De Morgan is its precursor (though his chief work was published
contemporaneously with Boole's in 1847); Boole set out the main
lines of the system in that year; but his exposition lacks the concept
of the logical sum which first appears in Peirce (1867), Schroder
(1877), and Jevons (1890), as also the concept of inclusion, originally
introduced by Gergonne (1816) and clearly formulated by Peirce in
1870. Schroder's system (1890) ranks as the completion of this
growth, though perhaps Peano's (1899) may here be counted as the
real close.
A. DE MORGAN
Boole's calculus emerged in a way from the 'classical' endeavours
to broaden the Aristotelian syllogistic (36.15 f.). This can be most
clearly seen from the syllogistic of Augustus de Morgan.
40.01 I shall now proceed to an enlarged view of the
proposition, and to the structure of a notation proper to repre-
sent its different cases.
As usual, let the universal affirmative be denoted by A,
the particular affirmative by /, the universal negative by E,
and the particular negative by O. This is the extent of the
common symbolic expression of propositions: I propose to
make the following additions for this work. Let one particular
choice of order, as to subject and predicate, be supposed
established as a standard of reference. As to the letters X, F,.
Z, let the order always be that of the alphabet, IF, YZ, XZ
Let x, y, z, be the contrary names of X, F, Z; and let the
same order be adopted in the standard of reference. Let the
four forms when choice is made of an X, F, Z, be denoted by
At, E , /,, O,; but when the choice is made from the contraries,
let them be denoted by A', E\ /', O' . Thus with reference to
Y and Z, "Every Y is Z" is the At of that pair and order:
while "Every y is z" is the A'. I should recommend ^and^'
296
BOOLEAN CALCULUS
to be called the sub-A and the super- A of the pair and order
in question: the helps which this will give the memory will
presently be very apparent. And the same of I t and /', etc.
Let the following abbreviations be employed; - X) Y
means "Every X is Y". X. Y means "No X is Y". X:Y means
"Some Xs are not Ys". XY means "Some Xs are Fs".
Later, De Morgan developed a different symbolism. We give it-
description and a comparative table, from a paper of 18.rjf):
40.02 Let the subject and predicate, when specified, be
written before and after the symbols of quantity. Let the
enclosing parenthesis, as in X) or (X, denote that the name-
symbol X, which would be enclosed if the oval were completed,
enters universally. Let an excluding parenthesis, as in )X
or X(, signify that the name-symbol enters particularly. Let
an even number of dots, or none at all, inserted between
the parentheses, denote affirmation or agreement; let an odd
number, usually one, denote negation or non-agreement.
40.03
Universals
Former
Notation of Both.
Notation
Proposition
memoir.
my Work on
now
expressed in
Logic.
proposed.
common
language.
A
A1 X)Y
X))Y
Every X is F
a
A1 x ) y or
x )) y or
Every F is X
Y)X
X(( Y
E
E1 X)yov
X))yov
No X is F
X . Y
X).( Y
e
E1 x ) For
x))Yor
Everything is
x.y
X(.) Y
X or F o rboth
Particulars
I
It XY
X()Y
Some Xs are Fs
i
I1 xy
x()yor
Some things
X)(Y
are neither .Ys
nor Fs
0
Ox Xy or
XQyor
Some Xs are
X : F
X (.( Y
not Fs
0
O1 xY or
x () F or
Some Fs are
F : X
X).)Y
not Xs
•297
MATHEMATICAL LOGIC
B. BOOLE
Boole, who was the first to outline clearly the program of mathe-
matical logic, was also the first to achieve a partial execution. In
this respect there is a great likeness between his relationship to
Leibniz and that of Aristotle to Plato. For with Boole as with
Aristotle we find not only ideas but a system.
This system of Boole's can be described thus: it is in the first
r^ace closely allied to arithmetic, in that it uses only arithmetical
symbols and has only one law that diverges from those of arith-
metic, viz. xn = x. All its procedures are taken over from simple
algebra; Boole has no conscious awareness of purely logical methods
(even of those which are intuitively used in algebra), e.g. of the rules
of detachment and substitution. As a matter of fact, even the basic
law mentioned makes very little difference to the algebraic character
of his system - which is algebra limited to the numbers 0 and 1.
Boole's mathematicism goes so far - and this is the second main
characteristic of his doctrine - that he introduces symbols and
procedures which admit of no logical interpretation, or only a com-
plicated and scarcely interesting one. Thus we meet with subtrac-
tion and division and numbers greater than 1.
From the logical point of view it is to be noted that disjunction
(symbolized by lx + y') is taken as exclusive, and that inclusion is
expressed by means of equality. Both lead to difficulties and
unnecessary complications; both are the result of the tendency
to mathematicize.
A third and special characteristic is that the system possesses
two interpretations, in classical and propositional logic.
Altogether, in spite of its defects, Boolean algebra is a very
successful piece of logic. Boole resembles Aristotle both in point of
originality and fruitfulness ; indeed it is hard to name another
logician, besides Frege, who has possessed these qualities to the
same degree, after the founder.
1. Symbolism and basic concepts
40.04 Proposition I. All the operations of the Language,
as an instrument of reasoning, may be conducted by a system
of signs composed of the following elements, viz. :
1st. Literal symbols as x, y, etc., representing things as
subjects of our conceptions.
2nd. Signs of operation, as +, -, x, standing for those
operations of the mind by which the conceptions of things
are combined or resolved so as to form new conceptions
involving the same elements.
3rd. The sign of identity, =.
298
BOOLEAN CALCULUS
And these symbols of Logic are in their use subject to
definite laws, partly agreeing with and partly differing from
the laws of the corresponding symbols in the science of
Algebra.
40.05 Let us employ the symbol 1 or unity, to represent
the Universe, and let us understand it as comprehending
every conceivable class of objects whether actually existing
or not, it being premised that the same individual may La
found in more than one class, inasmuch as it may possess
more than one quality in common with other individuals.
Let us employ the letters X, Y, Z, to represent the individual
members of classes, X applying to every member of one
class, as members of that particular class, and Y to every
member of another class as members of such class, and so on,
according to the received language of treatises on Logic.
Further let us conceive a class of symbols x, y, z, possessed
of the following character.
The symbol x operating upon any subject comprehending
individuals or classes, shall be supposed to select from that
subject all the Xs which it contains. In like manner the
symbol y, operating upon any subject, shall be supposed to
select from it all individuals of the class Y which are com-
prised in it, and so on.
When no subject is expressed, we shall suppose 1 (the
Universe) to be the subject understood, so that we shall
have
40.051 x = x (1),
the meaning of either term being the selection from the
Universe of all the Xs which it contains, and the result of
the operation being in common language, the class X, i.e. the
class of which each member is an X .
From these premises it will follow, that the product xy will
represent, in succession, the selection of the class Y, and the
selection from the class Y of such individuals of the class X
as are contained in it, the result being the class whose members
are both Xs and Fs. . . .
From the nature of the operation which the symbols x,
y, z, are conceived to represent, we shall designate them as
elective symbols. An expression in which they are involved
will be called an elective function, and an equation of which
the members are elective functions, will be termed an elective
equation. . . .
299
MATHEMATICAL LOGIC
1st. The result of an act of election is independent of the
grouping or classification of the subject.
Thus it is indifferent whether from a group of objects
considered as a whole, we select the class X, or whether we
divide the group into two parts, select the Xs from them
separately, and then connect the results in one aggregate
conception.
We may express this law mathematically by the equation
(40.052) x (u + v) = xu + xv,
u + v representing the undivided subject, and u and v the
component parts of it.
2nd. It is indifferent in what order two successive acts of
election are performed.
Whether from the class of animals we select sheep, and
from the sheep those which are horned, or whether from the
class of animals we select the horned, and from these such as
are sheep, the result is unaffected. In either case we arrive
at the class of horned sheep.
The symbolical expression of this law is
(40.053) xy = yx.
3rd. The result of a given act of election performed twice,
or any number of times in succession, is the result of the
same act performed once. . . . Thus we have
(40.054) xx = x,
or x2 = x:
and supposing the same operation to be n times performed,
we have
(40.055) xn =x,
which is the mathematical expression of the law above
stated.
The laws we have established under . . . symbolical forms . . .
are sufficient for the base of a Calculus. From the first of
these it appears that elective symbols are distributive, from
the second that they are commutative; properties which they
possess in common with symbols of quantity, and in virtue
of which, all the processes of common algebra are applicable
to the present system. The one and sufficient axiom involved
in this application is that equivalent operations performed
upon equivalent subjects produce equivalent results.
The third law ... we shall denominate the index law. It
is peculiar to elective symbols.
300
BOOLEAN CALCULUS
2. Applications
We now give two examples of the application of these principles
in Boole's work. The first concerns the law of contradiction.
40.06 That axiom of metaphysicians which is termed the
principle of contradiction, and which affirms that it is impos-
sible for any being to possess a quality, and at the same time
not to possess it, is a consequence of the fundamental law
of thought, whose expression is x2 = x.
Let us write this equation in the form
(40.061) x-x2 = 0
whence we have
(40.062) x(l -x) = 0; (1)
both these transformations being justified by the axiomatic
laws of contradiction and transposition. . . . Let us for simpli-
city of conception, give to the symbol x the particular inter-
pretation of men, then 1 - x will represent the class of 'not-
men'. . . . Now the formal product of the expressions of two
classes represents that class of individuals which is common
to them both. . . . Hence x (1 - x) will represent the class
whose members are at once 'men', and 'not-men', and the
equation (1) thus expresses the principle, that a class whose
members are at the same time men and not men does not exist.
In other words, that it is impossible for the same individual to
be at the same time a man and not a man. Now let the meaning
of the symbol x be extended from the representing of 'men',
to that of any class of beings characterized by the possession of
any quality whatever; and the equation (1) will then express
that it is impossible for a being to possess a quality and not
to possess that quality at the same time. But this is identically
that 'principle of contradiction' which Aristotle has de-
scribed as the fundamental axiom of all philosophy. . . .
The above interpretation has been introduced not on
account of its immediate value in the present system, but as
an illustration of a significant fact in the philosophy of the
intellectual powers, viz., that what has been commonly re-
garded as the fundamental axiom of metaphysics is but the
consequence of a law of thought, mathematical in its form.
The second example is taken from the application of Boole's
methods in the domain of syllogistic.
40.07 The equation by which we express any Proposition
concerning the classes X and Y, is an equation between the
301
MATHEMATICAL LOGIC
symbols x and y, and the equation by which we express any
proposition concerning the classes Y and Z, is an equation
between the symbols y and z. If from two such equations we
eliminate y, the result, if it do not vanish, will be an equation
between x and z, and will be interpretable into a Proposition
concerning the classes X and Z. And it will then constitute
the third member, or Conclusion, of a Syllogism, of which
the two given Propositions are the premises.
The result of the elimination of y from the equations
ay + b = 0,
(14)
a' y - b' = 0,
is the equation ab' - a' b = 0 (15).
40.08 Ex(ample). AA, Fig. 1, and by mutation of premises
(change of order), AA, Fig. 4.
All Ys are Xs, y (1 - x) = 0, or (1 - x) y = 0,
All Zs are Ys, z (1 - y) = 0, or zy - z = 0.
Eliminating y by (15) we have
z (1 - x) = 0,
All Zs are Xs.
In both these texts Boolean methods are being applied to tradi-
tional problems, involving logical relationships between two objects
(classes, propositions). But the interesting thing about this calculus
for our history is that it is applicable to more than two objects,
so that it oversteps the limits of the 'classical' logic. An instance
is given later (41.03).
C. THE LOGICAL SUM
The original Boolean calculus had two main defects from the
logical point of view, both occasioned by its extreme mathemati-
cism; disjunction was treated as exclusive, and there was no symbol
to hand for inclusion, though that is fundamental in logic. The first
defect was remedied by Jevons, who was strongly opposed to this
mathematicism and introduced non-exclusive disjunction.
49.09 There are no such operations as addition and sub-
traction in pure logic. . .
40.10 Now addition, subtraction, multiplication, and divi-
sion, are alike true as modes of reasoning in numbers, where
we have the logical condition of a unit as a constant restriction.
But addition and subtraction do not exist, and do not give
true results in pure logic, free from the conditions of number.
302
BOOLEAN CALCULUS
For instance take the logical proposition -
A+B+C=A+D+E
meaning what is either A or B or C is either A or D or E, and
vice versa. There being no exterior restriction of meaning
whatever, except that some terms must always have the same
meaning, we do not know which of A, D, E, is B, nor which
is C;. . . . The proposition alone gives us no such information.
Much clearer is Charles S. Peirce, also an opponent of Boole's
mathematicism (1867). He uses an appropriate though still primitive
symbolism.
40.11 Let the sign of equality with a comma beneath it
express numerical identity. . . . Let a -t b denote all the
individuals contained under a and b together. The operation
here performed will differ from arithmetical addition in two
respects : first, that it has reference to identity, not to equality,
and second, that what is common to a and b is not taken into
account twice over, as it would be in arithmetic. The first of
these differences, however, amounts to nothing, inasmuch as
the sign of identity would indicate the distinction in which it
is founded; and therefore we may say that
(1) If No a is b a -t 6 ^ a + b.
It is plain that
(2) a -t a f fl
and also, that the process denoted by -t, and which I shall call
the process of logical addition, is both commutative and
associative. That is to say
(3) a -b b =? b -t a
and
(4) (a h? b) -t c 7= a -t (b -t c).
This is the third time that non-exclusive disjunction is discovered,
cf. Galen (20.18) and Burleigh (30.20).
A symbolism quite different from that of mathematics is first met
with in Peano (41.20).
D. INCLUSION
The introduction of the concept of inclusion and a symbol for it
has a fairly long history. The modern symbol appears thirty years
before Boole's Analysis and quite independently of his calculus
in J. D. Gergonne's Essai de dialedique ralionelle, 1816/17. (The
parentheses enclosing the italic capitals in this text are Gergonne's.)
303
MATHEMATICAL LOGIC
40.12 We have chosen the signs to characterize these rela-
tions in the way which seems best for linking the sign to the
thing signified, and this is an endeavour which we think of
some importance, however puerile it may appear at first. The
letter (H), initial letter of the word Hors (outside) designates
the system of two ideas completely outside one another, as are
the two vertical strokes of this letter. These two strokes can
next be considered as crossed to form the letter (X) intended
to recall the system of two ideas which, as it were, somehow
intersect. Finally the two strokes can be identified so as to
form the letter (/) which we use to represent the system of
two ideas which exactly coincide with one another; this letter
is, moreover, the initial letter of the word Identity, the denomi-
nation suitable to the kind of relation in question. It may also
be noted that the three letters (H, X, I) are symmetrical,
like the relations they are intended to recall, so that they
are not liable to change their appearance by being reversed.
But this is not the case with the letter (67) which on being
reversed changes into (j); hence we have reserved this letter
to recall a relation in which the two ideas play different parts,
a relation which is not at all reciprocal. This letter is, moreover,
the initial letter common to both of the words Containing
and Contained, which well express the relative situation of the
two ideas.
But it was Charles S. Peirce who in 1870 systematically elaborated
the concept of inclusion.
40.13 Inclusion in or being as small as is a transitive relation.
The consequence holds that
If x-<y,
and y — < z,
then x — < z.
(Footnote) I use the sign — < in place of ^. My reasons for
not liking the latter sign are that it cannot be written rapidly
enough, and that it seems to represent the relation it expresses
as being compounded of two others which in reality are
complications of this. It is universally admitted that a higher
conception is logically more simple than a lower one under it.
Whence it follows from the relations of extension and com-
prehension, that in any state of information a broader concept
is more simple than a narrower one included under it. Now all
equality is inclusion in, but the converse is not true; hence
304
BOOLEAN CALCULUS
inclusion in is a wider concept than equality, and therefore
logically a simpler one. On the same principle, inclusion is also
simpler than being less than. The sign ^ seems to involve a
definition by enumeration; and such a definition offends
against the laws of definition.
Schroder introduces and explains the symbol of inclusion from
the start:
40.14 Examples of categorical judgements of the simplest
kind are propositions accepted as true in chemistry:
'Gold is metal' - 'Common salt is sodium chloride'. -
Even to these we can very easily link the basic contrasts
needed in our science.
Both statements have the same copula. . . . Yet the factual
relation between the subject and predicate of the statement is
essentially different in the first and in the second case, insofar
as conversely metal is not always gold, while, sodium chloride is
also common salt. This difference is not expressed in a way
apparent to the eye in the original statements.
If it is now desired to exhibit the factual relation between
subject and predicate by a relative symbol more exactly than
those statements do, a symbol must be chosen for the first
example different from that for the second. One might write :
gold (^ metal common salt = sodium chloride
40.15 The other symbol Q can be read . . . 'subordinated'. It
is called the symbol of subordination and a statement such as
aQb,
a 'subordination'. The symbol is shaped similarly to, and to
some extent in imitation of, the 'inequality symbol' of
arithmetic, viz. the symbol < for 'less [than]'. As is well
known, this can be read backwards as 'greater', >, and it is
easily impressed on the memory together with its meaning if
one bears in mind that the symbol broadens from the smaller
to the larger value, or points from the larger value towards
the smaller. Analogously, our symbol of subordination, when
read backwards in the reversed position, ^), i.e. reading again
from left to right, will mean ' super ordinate d\ The original
subordination may also be written backwards as a superordi-
nation' :
and this expression means just the same as the original one.
305
MATHEMATICAL VARIETY OF LOGIC
40.16 The copula 'is' is sometimes used to express one,
sometimes the other of the relations which we have shown
by means of the symbols (2 and =. For its exhibition a symbol
composed of both the two last, =£, is chiefly recommended, as
being immediately, and so to say of itself, intelligible, and
readily memorizable. In fullest detail, this symbol is to be
read as 'subordinated or equaV . . . .
A statement of the form
a=C°
is called a sub sumption, the symbol =£[ the symbol of sub-
sumption.
E. PEANO
The term of this whole development is to be found in the sym-
bolism which Giuseppe Peano published in 1889. This comprises
essentially more than the Boolean calculus and at the same time
brings the latter to its final form. Its essentials will be given below
(41.20).
306
III. PROPOSITIONAL LOGIC
§41. PROPOSITIONAL LOGIC: BASIC CONCEPTS
AND SYMBOLISM
We speak first of the development of proposition-determining
functors and other fundamental parts of proposition;! I Logic. This
was first formulated, in the modern period of Logic, by Boole -
actually as the second possible interpretation of his calculus (1847 .
A more exact exposition appears in McColl (1877). Frege's Begrif/s-
schrift (1879) marks a new beginning, in this as in so many other
regions of formal logic. In connection with Frege's doctrine of
implication we give also two important texts from Peirce.
Later, Peano (1889) introduced a symbolism which is notably
easier to read than Frege's; Russell's displays only inessential
variations from it. But the symbolism which Lukasiewicz later
constructed, in dependence on Frege, is basically different from
Peano's.
A. BOOLE
We read in the Analysis:
41.01 Of the conditional syllogism there are two, and only
two formulae.
1st The constructive,
If A is B, then C is D,
But A is B, therefore C is D.
2nd The destructive,
If A is B, then C is D,
But C is not D, therefore A is not B.
... If we examine either of the forms of conditional syllog-
ism above given, we shall see that the validity of the argument
does not depend upon any considerations which have reference
to the terms A, B, C, D, considered as the representatives of
individuals or of classes. We may, in fact, represent the
Propositions A is B, C is D, by the arbitrary symbols X and Y
respectively, and express our syllogisms in such forms as the
following :
If X is true, then Y is true,
But X is true, therefore Y is true.
Thus, what we have to consider is not objects and classes
of objects, but the truths of Propositions, namely, of those
elementary Propositions which are embodied in the terms of
our hypothetical premises.
307
MATHEMATICAL VARIETY OF LOGIC
41.02 If we confine ourselves to the contemplation of a
given proposition X, and hold in abeyance every other conside-
ration, then two cases only are conceivable, viz. first that the
given Proposition is true, and secondly that it is false. As
these cases together make up the Universe of the Proposition,
and as the former is determined by the elective symbol x,
the latter is determined by the symbol 1 - x.
But if other considerations are admitted, each of these
cases will be resolvable into others, individually less extensive,
the number of which will depend upon the number of foreign
considerations admitted. Thus if we associate the Propositions
X and y, the total number of conceivable cases will be found
as exhibited in the following scheme.
Cases Elective expressions
1st X true, y true xy
2nd X true, Y false x (1 -y)
3rd X false, Y true (1 -x) y
4th X false, Y false (l-x){l-y).
41.03 And it is to be noted that however few or many those
circumstances may be, the sum of the elective expressions
representing every conceivable case will be unity. Thus let
us consider the three Propositions. X, It rains, Y, It hails,
S, It freezes. The possible cases are the following:
Cases Elective expressions
1st It rains, hails, and freezes, xyz
2nd It rains and hails, but does not
freeze xy (1 - z)
3rd It rains and freezes, but does not
hail xz (1 - y)
4th It freezes and hails, but does not yz (1 - x)
rain
5th It rains, but neither hails nor x (1 - y) (1 - z)
freezes
6th It hails, but neither rains nor
freezes y (1 -x) (1 - z)
7th It freezes, but neither hails nor z (1 - x) (1 - y)
rains
8th It neither rains, hails, nor freezes (1 - x) (1 - y) (1 - z)
1 — sum
308
PROPOSITIONAL LOGIC
41.04 ... To express that a given Proposition X is true.
The symbol 1 - x selects those cases in which the Propo-
sition X is false. But if the Proposition is true, there are no
such cases in its hypothetical Universe, therefore
1 - x = 0,
or x = 1.
To express that a given Proposition X is false.
The elective symbol x selects all those cases in which the
Proposition is true, and therefore if the Proposition is false,
x = 6.
These principles are then applied just like those of the logic of
classes, to syllogistic practice.
The similarity of the table of four cases in 41.02 with Philo's
matrix of truth-values (20.07) is to be noticed. As has already been
said, the Boolean calculus had no symbol for implication, nor yet
one for negation; both are introduced by means of more complex
formulae. In place of the logical sum, Boole had the notion of
exclusive disjunction. Hence it is that propositional logic is made
to appear as a discipline co-ordinate with, if not subordinate to,
the logic of classes, by contrast to the clear insight possessed by the
Stoics and Scholastics into its nature as basic.
B. McCOLL
Passing over the development that occurred between 1847 and
1877, mainly due to Jevons and Peirce, we now give instead a text
from Hugh McGoll (1877) in which propositional logic is emancipated
from the calculus of classes, and endowed with all the symbols just
mentioned. In a way this text marks the highest level of mathe-
matical logic before Frege.
41.05 Definition 1. - Let any symbols, say A, B, C, etc.,
denote statements [or propositions] registered for con-
venience of reference in a table. Then the equation A = 1
asserts that the statement A is true; the equation A = 0
asserts that the statement A is false ; and the equation A = B
asserts that A and B are equivalent statements.
41.06 Definition 2. - The symbol A x B x C or ABC
denotes a compound statement, of which the statements
A, B, C may be called the factors. The equation ABC = 1
asserts that all the three statements are true; the equation
ABC = 0 asserts that all the three statements are not true,
i.e. that at least one of the three is false. Similarly a com-
pound statement of any number of factors may be defined.
309
MATHEMATICAL VARIETY OF LOGIC
41.07 Definition 3. - The symbol A + B + C denotes an
indeterminate statement, of which the statements A, B, C
may be called the terms. The equation A + B + C = 0 asserts
that all the three statements are false ; the equation A + B + C
= 1 asserts that all the three statements are not false, i.e.,
that at least one of the three is true. Similarly an indeterminate
statement of any number of terms may be defined.
41.08 Definition 4. - The symbol A' is the denial of the
statement A. The two statements A and A' are so related
that they satisfy the two equations A + A' = 1 and A A ' = 0;
that is to say, one of the two statements (either A or A') must
be true and the other false. The same symbol {i.e. a dash)
will convert any complex statement into its denial. For
example, (AB)' is the denial of the compound statement
AB. . . .
41.09 Definition 5. - When only one of the terms of an
indeterminate statement A + B + C + . . . can be true, or
when no two terms can be true at the same time, the terms
are said to be mutually inconsistent or mutually exclusive.
41.10 Definition 12. - The symbol A: B [which may be call-
ed an implication] asserts that the statement A implies B; or
that whenever A is true B is also true.
Note. - It is evident that the implication A:B and the
equation A = AB are equivalent statements.
C. FREGE
1. Content and judgment
A new period of propositional logic begins with Gottlob Frege.
His first work, the Begriffsschrift of 1879, already contains in brief
an unusually clear and thorough presentation of a long series of
intuitions unknown to his immediate predecessors, while those
already familiar are better formulated. To start with we choose a
text relevant rather to semantics than logic, in which this great
thinker introduces of his propositional logic with the 'judgment-
stroke' :
41.11 A judgment is always to be expressed by means of
the sign
This stands to the left of the sign or complex of signs in which
the content of the judgment is given. If we omit the little
vertical stroke at the left end of the horizontal stroke, then
310
PROPOSITIONAL LOGIC
the judgment is to be transformed into a mere complex of
ideas; the author is not expressing his recognition or non-
recognition of the truth of this. Thus, let
I— A
mean the judgment: 'unlike magnetic poles attract one
another'. In that case
A
will not express this judgment; it will be intended just to
produce in the reader the idea of the mutual attraction of
unlike magnetic poles - so that, e.g., he may make inferences
from this thought and test its correctness on the basis of
these. In this case we qualify the expression with the words
'the circumstance thaf or 'the proposition lhal\
Not every content can be turned into a judgment by
prefixing | — to a symbol for the content; e.g., the idea 'house'
cannot. Hence we distinguish contents that are, and contents
that are not, possible contents of judgment.
As a constituent of the sign | — the horizontal stroke combines
the symbols following it into a whole; assertion, which is expressed
by the vertical stroke at the left end of the horizontal one, relates
to the whole thus formed. The horizontal stroke I wish to call
the content-stroke, and the vertical the judgment-stroke. The
content-stroke is also to serve the purpose of relating any sign
whatsoever to the whole formed by the symbols following
the stroke. The content of what follows the content-stroke must
always be a possible content of judgment.
2. Implication
Frege then introduces the Philonian concept of implication,
though, unlike Peirce (41.14) he knows nothing in this connection
of Philo or the Scholastics. It is remarkable that he proceeds almost
exactly like Philo.
41.12 If A and B stand for possible contents of judgment,
we have the four following possibilities :
(i) A affirmed, B affirmed;
(ii) A affirmed, B denied;
(iii) A denied, B affirmed;
(iv) A denied, B denied.
l— B
stands for the judgment that the third possibility is not
311
MATHEMATICAL VARIETY OF LOGIC
realized, but one of the other three is. Accordingly, the denial of
l— B
is an assertion that the third possibility is realized, i.e. that
A is denied and B affirmed.
From among the cases where
•— B
is affirmed, the following may be specially emphasized:
(1) A is to be affirmed. - In this case the content of B is
quite indifferent. Thus, let i — A mean: 3 x 7 = 31 ; let £
stand for the circumstance of the sun's shining. Here only
the first two cases out of the four mentioned above are
possible. A causal connection need not exist between the two
contents.
(2) B is to be denied. - In this case the content of A is
indifferent. E.g., let B stand for the circumstance of perpetual
motion's being possible, and A for the circumstance of the
world's being infinite. Here only the second and fourth of the
four cases are possible. A causal connection between A and B
need not exist.
(3) One may form the judgment
l— B
without knowing whether A and B are to be affirmed or denied.
E.g., let B stand for the circumstance of the Moon's being in
quadrature with the Sun, and A the circumstance of her
appearing semi-circular. In this case we may render
l— B
by means of the conjunction 'if; 'if the Moon is in quadrature
with the Sun, then she appears semi-circular'. The causal
connection implicit in the word 'if is, however, not expressed
by our symbolism; although a judgment of this sort can be
made only on the ground of such a connection. For this con-
nection is something general, and as yet we have no expression
for generality.
The text needs some explanations. First, Frege uses A1 for the
consequent and ' B' for the antecedent - contrary to ordinary usage,
but like Aristotle; so the antecedent stands in the lower place. So
312
PROPOSITIONAL LOGIC
the schema excludes only the case where the antecedent (B) is true
and the consequent (A) is false; in all the other three cases the
proposition is true. Thus we have just the same state of affairs as in
Philo (20.07): the schema is a symbol of Philonian implication. It
signifies 'if B, then A1 in the Philonian sense of 'if.
Important is the stress laid on the fact that implication has
nothing to do with the causal connection between the facts signified
by the antecedent and consequent. .
D. PEIRCE
Philonian implication alone continued to be used in mathematical
logic up to 1918 - unlike usage in the Stoic and Scholastic periods.
One of the best justifications of this concept which seems so odd to
the man in the street, is to be found in a fairly late text of Peirce's,
dated 1902.
41.13 To make the matter clear, it will be well to begin by
defining the meaning of a hypothetical proposition, in general.
What the usages of language may be does not concern us;
language has its meaning modified in technical logical for-
mulae as in other special kinds of discourse. The question
is what is the sense which is most usefully attached to the
hypothetical proposition in logic ? Now, the peculiarity of the
hypothetical proposition is that it goes out beyond the actual
state of things and declares what would happen were things
other than they are or may be. The utility of this is that
it puts us in possession of a rule, say that 'if A is true, B is
true', such that should we hereafter learn something of which
we are now ignorant, namely that A is true, then by virtue
of this rule, we shall find that we know something else, namely,
that B is true. There can be no doubt that the Possible, in its
primary meaning, is that which may be true for aught we
know, that whose falsity we do not know. The purpose is
subserved, then, if throughout the wrhole range of possibility,
in every state of things in which A is true, B is true too.
The hypothetical proposition may therefore be falsified by a
single state of things, but only by one in which A is true
while B is false. States of things in which A is false, as well
as those in which B is true, cannot falsify it. If, then, B is a
proposition true in every case throughout the whole range of
possibility, the hypothetical proposition, taken in its logical
sense, ought to be regarded as true, whatever may be the
313
MATHEMATICAL VARIETY OF LOGIC
usage of ordinary speech. If, on the other hand, A is in no
case true, throughout the range of possibility, it is a matter
of indifference whether the hypothetical be understood to
be true or not, since it is useless. But it will be more simple
to class it among true propositions, because the cases in which
the antecedent is false do not, in any other case, falsify a
hypothetical. This, at any rate, is the meaning which I shall
attach to the hypothetical proposition in general, in this
paper.
Also of interest is the following remark of the same logician
(1896):
41.14 Although the Philonian views lead to such incon-
veniences as that it is true, as a consequence de inesse, that if
the Devil were elected president of the United States, it
would prove highly conducive to the spiritual welfare of the
people (because he will not be elected), yet both Professor
Schroder and I prefer to build the algebra of relatives upon
this conception of the conditional proposition. The incon-
venience, after all, ceases to seem important, when we reflect
that, no matter what the conditional proposition be under-
stood to mean, it can always be expressed by a complexus of
Philonian conditionals and denials of conditionals.
E. APPLICATIONS OF HIS SYMBOLISM BY FREGE
Some examples of the applications of Frege's implication-schema
will make his main ideas clearer.
41.15 The vertical stroke joining the two horizontal ones
is to be called the conditional-stroke. . . . Hence it is easy to
see that
I rA
]— B
— r
denies the case in which A is denied, B and V are affirmed.
This must be thought of as compounded of
,4 and/1
just as
314
PROPOSITIONAL LOGIC
"TZb
is from A and B. Thus we first have the denial of the case
in which
is denied, Y is affirmed. But the denial of
signifies that A is denied and B is affirmed. Thus we obtain
what is given above.
41.16 From the explanation given in § 5 (41.12) it is obvious
that from the two judgments
| — , — A and I — B
\—B
there follows the new judgment i — A. Of the four cases
enumerated above, the third is excluded by
\—B
and the second and fourth by :
\— B,
so that only the first remains.
41.17 Let now X for example signify the judgment
I— B
— or one which j — , — A contains as a particular case. Then
I— B
I write the inference thus:
\— B
(X):
I— A.
Here it is left to the reader to put together the judgment
]— B
from | — B and | — A, and see that it tallies with the cited
judgment X.
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MATHEMATICAL VARIETY OF LOGIC
F. NEGATION AND SUM IN FREGE
Frege uses the same schemata, together with the 'negation-stroke',
to express the logical sum.
41.18 If a small vertical stroke is attached to the lower
side of the content-stroke, this shall express the circumstance
of the content's not being the case. Thus, e.g., the meaning
of
h-r- A:
is: 'A is not the case'. I call this small vertical stroke the
negation-stroke.
41.19 We now deal with some cases where the symbols
of conditionality and negation are combined.
\—B
means : 'the case in which B is to be affirmed and the negation
of A is to be denied does not occur'; in other words, 'the
possibility of affirming both A and B does not exist', or A and
B are mutually exclusive'. Thus only the three following
cases remain:
A affirmed, B denied;
A denied, B affirmed;
A denied, B denied.
From what has already been said, it is easy to determine
the meaning possessed by each of the three parts of the
horizontal stroke preceding A.
means: 'the case in which A is denied and negation of B is
affirmed does not exist'; or, lA and B cannot both be denied'.
There remain only the following possibilities :
A affirmed, B affirmed;
A affirmed, B denied;
A denied, B affirmed.
A and B between them exhaust all possibilities. Now the
words 'or', 'either - or', are used in two ways. In its first
meaning,
means just the same as
316
A or B'
A
B,
PROPOSITIONAL LOGIC
i. e. that nothing besides A and B is thinkable. E.g., if a
gaseous mass is heated, then either its volume or its pressure
increases. Secondly, the expression
lA or B'
may combine the meaning of
1 F" A and that of [~T~ A
— B 4" B
so that (i) there is no third possibility besides A and B, (ii) A
and B are mutually exclusive. In that case only the following
two possibilities remain out of the four:
A affirmed, B denied;
A denied, B affirmed.
Of these two uses of the expression 'A or J3' the more impor-
tant is the first, which does not exclude the coexistence of A
and B; and we shall use the word lor' with this meaning. Perhaps
it is suitable to distinguish between 'or' and 'either - or',
regarding only the latter as having the subsidiary meaning
of mutual exclusion.
G.PEANO'S SYMBOLISM FOR PROPOSITIONAL LOGIC
Frege's symbolism has the unusual feature of being two-dimen-
sional. In that it diverges from the historical practice of mankind
which has almost always expressed its thoughts in one-dimensional
writing. It must be admitted that this revolutionary novelty has
much to be said for it - it notably widens the expressive possibilities
of writing. But this was too revolutionary; Frege's symbolism did
not prove generally intelligible, and the subsequent development took
place in another direction. Schroder made no reference to it in 1892,
Russell admitted in 1903 that he had learned much from Frege when
he had met his system, but not having known it he followed Peano.
Modern mathematical logic, though its authors have less depth of
thought than Frege, has adopted Peano's symbolism. For this
reason we quote a text from Peano's Arithmetices Principia (1889) in
which he lays down this intuitively clear and meaningful symbolism
for propositional logic.
41.20 I. Concerning punctuation
By the letters a, b, ... x, y, ... x', y' ... we indicate any
undetermined beings. Determined beings we indicate by the
signs or letters P, K, N. . . .
For the most part we shall write signs on one and the same
line. To make clear the order in which they are to be con-
joined we use parentheses as in algebra, or dots .:/.:: etc.
317
MATHEMATICAL VARIETY OF LOGIC
That a formula divided by dots may be understood, first
the signs which are separated by no dots are to be collected,
afterwards those separated by one dot, then those by two
dots, etc.
E.g. let a, 6, c, . . . be any signs. Then ab • cd signifies (ab)(cd) ;
and ab . cd : ef . gh .-. k signifies (((ab) (cd)) ((ef) (gh))) k.
Signs of punctuation may be omitted if there are formulae
with different punctuation but the same sense; or if only
one formula, and that the one we wish to write, has the
sense.
To avoid danger of ambiguity we never make use of . : as
signs of arithmetical operations.
The only form of parentheses is (). If dots and paren-
theses occur in the same formula, signs contained in paren-
theses are to be collected first.
//. Concerning propositions
By the sign P is signified a proposition.
The sign r» is read and. If a, b are propositions; then a r» b
is the simultaneous affirmation of the propositions a, b. For
the sake of brevity we shall commonly write ab in place
of a n b.
The sign - is read not. Let abeaP; then -a is the negation
of the proposition a.
The sign w is read or (vet). Let a, b be propositions; then
a v b is the same as - \-a.-b.
[By the sign V is signified verum or identity; but we never
use this sign.]
The sign A signifies falsum or absurdum.
[The sign C signifies is a consequence; thus bCa is read b
is a consequence of the proposition a. But we never use this
sign.]
The sign j signifies is deduced (deducitur) : thus a j b signi-
fies the same as bCa.
H. LATER DEVELOPMENT OF SYMBOLISM
FOR PROPOSITIONAL LOGIC
Peano's successors introduced only minor alterations to his
symbolism. First Russell (1903), who writes 'v' instead of 'w', and
l~ instead of '-', then Hilbert and Ackermann (1928) who write
a stroke over a letter for negation, and 'oo' for the equivalence-
318
PROPOSITIONAL LOGIC
sign ' = ' of Frege and Russell. M. H. ShefTer (1928) introduced '|' as a
sign for 'not both'.
The Polish school, on the other hand, developed two symbolic
languages essentially different from Peano's; those of St. Lesniewski
and J. Lukasiewicz. We shall not go into the first, which is peculiar
and little used, but the symbolism of Lukasiewicz deserves brief
exposition, both for its originality and its exactness. The essential
feature is that all predicates (called by Lukasiewicz 'functors';
stand in front of their arguments; thus all brackets and dots are
dispensed with, without any ambiguity arising.
The various sets of symbols may be compared thus:
McColl A' + x : ' =
Peano
-P
w
r\
D
=
Russell
r+~>p
V
.
D
=
Hilbert
A
&
->
CO
Lukasiewicz
Np
A
K
c
E
D [i.e.NK)
Thus Lukasiewicz writes 'Cpq' for 'p d q\ and lApq' for 'p v q' .
An example of a more complex formula is 'CCpqCNqNp' instead
of 'p D q . D . ^ q D r*** p\
§42. FUNCTION, VARIABLE, TRUTH-VALUE
While nearly everything mentioned so far is within the scope of
Stoic and Scholastic logic - though the new logicians knew hardly
anything of the achievements of their predecessors, - the concepts of
function, variable, and truth-value, without effecting anything radi-
cally new, yet produce so marked a development of the old concept
of logical form as to deserve distinct and thorough treatment.
After an introductory quotation from De Morgan (1858) con-
cerning logical form in general, we give Frege's fundamental text on
the concept of function (1893), the explanation and development of
Frege's thought from Russell and the Principia (1903 and 1910),
and finally the extension of the concept of function to many-place
functions by Peirce (1892) and Frege (1893). For the doctrine of
the variable in mathematical logic we have a quotation from Frege's
Begriffsschrift (1879) and Russell's elaboration of the ideas therein
(1903 and 1910). As to truth-values, two texts from Frege (1893)
and one from Peirce (1885) are to hand.
In conclusion we exhibit some examples of modern truth-matrices
(truth-tables, tables of truth-values), by which propositional func-
tors are defined, taking these from Peirce (1902) and Wittgenstein
(1921); the decision procedure based on them is illustrated from
Kotarbinski (1929).
319
MATHEMATICAL VARIETY OF LOGIC
A. LOGICAL FORM
An important text of De Morgan's make a fitting start. It dates
from 1858, and shows a very clear idea of logical form. It may be
compared with Buridan's definition of logical form (26.12): the
thought is the same, but more developed, in that abstraction is made
from the sense of the logical constants.
42.01 In the following chain of propositions, there is
exclusion of matter, form being preserved, at every step: -
Hypothesis
(Positively true) Every man is animal
Every man is Y Y has existence
Every X is Y X has existence
Every X Y is a transitive
relation
a of X Y a a fraction < or = 1
(Probability p) (3 of X Y pa fraction < or = 1
The last is nearly the purely formal judgment, with not a
single material point about it, except the transitiveness of the
copula. But 'is' is more intense than the symbol , which
means only transitive copula: for 'is' has transitiveness, and
more. Strike out the word transitive, and the last line shews the
pure form of the judgment.
The foregoing table is to be understood in the sense that the
conditions formulated in one row, hold for all subsequent rows; thus,
e.g., the relation shown by the stroke (' ') in the two last rows
must be transitive, since this is laid down in the preceding row.
Neither De Morgan nor any other logician can remain at so high
a level of abstraction as is here achieved. Basically, this is a re-
discovery of the scholastic concept of form, made through a broaden-
ing of the mathematical concept of function, for which we refer to
Peirce (42.02) and Frege.
B. CONCEPT OF FUNCTION: FREGE
We now give Frege's fundamental text (1893):
42.03 If we are asked to state what the word 'function' as
used in mathematics originally stood for, we easily fall into
saying that a function of x is an expression formed, by means
of the notations for sum, product, power, difference, and so
on, out of V and definite numbers. This attempt at a
definition is not successful because a function is here said to
320
PROPOSITIONAL LOGIC
be an expression, a combination of signs, and not what the
combination designates. Accordingly another attempt would
be made: we could try 'reference of an expression' instead of
'expression'. There now appears the letter V which indicates
a number, not as the sign '2' does, but indefinitely. For
different numerals which we put in the place of V, we get, in
general, a different reference. Suppose, e.g., that in the
expression '(2 + 3 • x2) x\ instead of '#' we put the number-
signs '0', '1', '2', '3', one after the other; we then get corres-
pondingly as the reference of the expression the numbers
0, 5, 28, 87. Not one of the numbers so referred to can claim
to be our function. The essence of the function comes out
rather in the correspondence established between the numbers
whose signs we put for V and the numbers which then appear
as the reference of our expression - a correspondence which is
represented intuitively by the course of the curve whose
equation is, in rectangular co-ordinates, 'y = (2 + 3. x2) x\ In
general, then, the essence of the function lies in the part of
the expression which is there over and above the lx\ The
expression of a function needs completion, is 'unsaturated' . The
letter lx' only serves to keep places open for a numerical sign
to be put in and complete the expression; and thus it enables
us to recognize the special kind of need for a completion that
constitutes the peculiar nature of the function symbolized
above. In what follows, the Greek letter %' will be used
instead of the letter V. This 'keeping open' is to be understood
in this way: All places in which '£' stands must always be
filled by the same sign and never by different ones. I call these
places argument-places, and that whose sign or name takes
these places in a given case I call the argument of the function
for this case. The function is completed by the argument:
I call what it becomes on completion the value of the function
for the argument. We thus get a name of the value of a func-
tion for an argument when we fill the argument-places in the
name of the function with the name of the argument. Thus,
e.g., '(2 + 3 . I2) 1' is name of the number 5, composed of the
function-name '(2 + 3 . £2) £' and '1'. The argument is not to
be reckoned in with the function, but serves to complete the
function, which is 'unsaturated' by itself. When in the sequel
an expression like 'the function O (£)' is used, it is always to be
observed that the only service rendered by '£' in the symbol
for the function is that it makes the argument-places recogniz-
321
MATHEMATICAL VARIETY OF LOGIC
able; it does not imply that the essence of the function
becomes changed when any other sign is substituted for '£'.
The following remarks will assist understanding of this pioneer
passage. In mathematical usage the word 'function' has two refer-
ences, usually not very clearly distinguished. On the one hand it
stands for an expression (formula) in which a variable occurs, on the
other for the 'correspondence between numbers' for which such an
expression stands, and so for some kind of lecton or in general, for
that for which the expression stands, (which in any case is not a
written symbol). Frege makes a sharp distinction between these two
references, and allows only the second to the word 'function' -
conformably with his general position that logic (and mathematics)
has as its object not symbols but what they stand for. It is important
to understand this, because Russell and nearly all logicians after him
will speak of expressions and formulae as 'functions', unlike Frege.
However, this opposition is irrelevant to the basic logical problems
considered here. Frege, too, makes use of analysis of expressions to
convey his thought, and what he states in the text just quoted,
holds good for every interpretation of the word 'function'. He intro-
duces, namely, three fundamental concepts: 1. of the argument and
argument-place, 2. of a value, 3. of an 'unsaturated' function, i.e.
one containing a variable.
C. PROPOSITIONAL FUNCTION: RUSSELL
Russell who knew the work of Frege well, followed his ideas but
with some divergences. He seems to start from the Aristotelian
concept of proposition rather than from the mathematical concept
of function, and as already said, apparently interprets the word
'function' as the name of an expression or written formula. In the
Principles (1903) he writes:
42.04 It has always been customary to divide propositions
into subject and predicate; but this division has the defect
of omitting the verb. It is true that a graceful concession is
sometimes made by loose talk about the copula, but the verb
deserves far more respect than is thus paid to it. We may say,
broadly, that every proposition may be divided, some in
only one way, some in several ways, into a term (the subject)
and something which is said about the subject, which some-
thing I shall call the assertion. Thus 'Socrates is a man' may
be divided into Socrates and is a man. The verb, which is the
distinguishing mark of propositions, remains with the asser-
tion; but the assertion itself, being robbed of its subject, is
neither true nor false. . . .
322
PROPOSITIONAL LOGIC
If this text is compared with 12.01 and similar passages in Aris-
totle, it can be seen that Russell here opts for the original Aristotelian
analysis of propositions against that of the later 'classical' logic.
In this connection he seems to have been the first to formulate
expressly the idea that when the subject is replaced by a variable, the
resulting formula - the propositional function is no longer a
proposition. The same problem is still more explicitly treated in the
Principia:
42.05 By a 'propositional function' we mean something
which contains a variable x, and expresses a proposition as
soon as a value is assigned to x. That is to say, it differs from
a proposition solely by the fact that it is ambiguous: it
contains a variable of which the value is unassigned. . . .
42.06 The question as to the nature of a function is by no
means an easy one. It would seem, however, that the essential
characteristic of a function is ambiguity. Take, for example,
the law of identity in the form lA is A\ which is the form in
which it is usually enunciated. It is plain that, regarded
psychologically, we have here a single judgment. But what
are we to say of the object of judgment? We are not judging
that Socrates is Socrates, nor that Plato is Plato, nor any
other of the definite judgments that are instances of the law
of identity. Yet each of these judgments is, in a sense, within
the scope of our judgment. We are in fact judging an ambi-
guous instance of the propositional function 'A is A'. We appear
to have a single thought which does not have a definite
object, but has as its object an undetermined one of the values
of the function 'A is A'. It is this kind of ambiguity that
constitutes the essence of a function. When we speak of lyx',
where x is not specified, we mean one value of the function,
but not a definite one. We may express this by saying that 'yx'
ambiguously denotes cpa, 96, cpc, etc., where cpa, 96, 9c, etc.
are the various values of lyx'.
D. MANY-PLACE FUNCTIONS
Perhaps even more important than the broadening of the concept
of function to include non-mathematical domains, is the extension to
many-place functions achieved by Frege and Peirce. The resulting
extension of the Aristotelian subject-predicate schema is something
quite new in formal logic. Our first text is Peirce's (1892) :
42.07 If upon a diagram we mark two or more points to
be identified at some future time with objects in nature, so as
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MATHEMATICAL VARIETY OF LOGIC
to give the diagram at that future time its meaning; or if in
any written statement we put dashes in place of two or more
demonstratives or pro-demonstratives, the professedly incom-
plete representation resulting may be termed a relative rhema.
It differs from a relative term only in retaining the 'copula', or
signal of assertion. If only one demonstrative or pro-demon-
strative is erased, the result is a non-relative rhema. For
example, ' buys from for the price of ', is a
relative rhema; it differs in a merely secondary way from
' is bought by from for ' ,
from ' sells --- to for ',
and from ' is paid by to for '.
On the other hand, ' is mortal' is a non-relative rhema.
42.08 A rhema is somewhat closely analogous to a chemical
atom or radicle with unsaturated bonds. A non-relative
rhema is like a univalent radicle; it has but one unsaturated
bond. A relative rhema is like a multivalent radicle. The
blanks of a rhema can only be filled by terms, or, what is the
same thing, by 'something which' (or the like) followed by a
rhema; or, two can be filled together by means of 'itself or
the like. So, in chemistry, unsaturated bonds can only be
saturated by joining two of them, which will usually, though
not necessarily, belong to different radicles. If two univalent
radicles are united, the result is a saturated compound. So,
two non-relative rhemas being joined give a complete propo-
sition. Thus, to join ' is mortal' and 'is a man', we
have 'A is mortal and AT is a man', or some man is mortal. So
likewise, a saturated compound may result from joining two
bonds of a bivalent radicle; and, in the same way, the two
blanks of a dual rhema may be joined to make a complete
proposition. Thus, ' loves ', 'A loves A', or something
loves itself.
Frege, a year later, writes in the same sense:
42.09 So far we have only spoken of functions of one
argument; but we can easily make the transition to func-
tions with two arguments. These need a double completion in
that after a completion by one argument has been effected, a
function with one argument is obtained. Only after another
completion do we reach an object, which is then called the
value of the function for the two arguments. Just as we made
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PROPOSITIONAL LOGIC
use of the letter '£' for functions with one argument, so we now
use the letters '£' and %' to express the twofold unsaturatedness
of functions with two arguments, as in '(<; + £)2 + £\ In
substituting, e.g., '1' for '£' we saturate the function to the
extent that in (§ + l)2 + 1 we are left with a function with
only one argument. This way of using the letters '£' and '£'
must always be kept in view when an expression such as 'the
function T (£, £)' occurs (cf. 42.03). ... I call the places in
which '£' appears, ^-argument-places, and those in which '£'
appears, ^-argument- places. I say that the ^-argument-
places are mutually cognate and similarly the ^-argument-
places, while I call a ^-argument-place not cognate to a £-
argument-place.
The functions with two arguments \ = £ and \ > £ always
have a truth-value as value [at least when the signs '='
and '>' are appropriately explained]. For our purposes we
shall call such functions relations. E.g., 1 stands to 1 in the
first relation, and generally every object to itself, while 2
stands to 1 in the second relation. We say that the object T
stands to the object A in the relation T (£, ?) if Y (I\ A) is the
True. Similarly we say that the object A falls under the con-
cept O (£), if O (A) is the True. It is naturally presupposed
here, that the function O (£) always has a truth-value. (Foot-
note of Frege' s : A difficulty occurs which can easily obscure the
true state of affairs and so cast doubt on the correctness of
my conception. When we compare the expression 'the truth-
value of this, that A falls under the concept O (£)' with 'O
(A)', we see that to '® ( )' there properly corresponds 'the
truth-value of this, that ( ) falls under the concept O (£)' and
not 'the concept O (£)'. Thus the last words do not properly
signify a concept [in our sense], though the form of speech
makes it seem as if they do. As to the difficulty in which
language thus finds itself, cf. my paper On Concept and Object.)
E. THE VARIABLE
1. Frege
Variables, introduced by Aristotle, were subsequently regularly
used both in logic and mathematics. A reflective concept of variable
is already to be found in Alexander of Aphrodisias (24.08V In
mathematical logic the concept of variable is first explicitly intro-
duced by Frege.
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MATHEMATICAL VARIETY OF LOGIC
42.10 The symbols used in the general theory of magnitude
fall into two kinds. The first consists of the letters ; each letter
represents either an indeterminate number or an indeter-
minate function. This indeterminateness makes it possible to
express by means of letters the general validity of propositions ;
e.g.: [a + b) c = ac + be. The other kind contains such
symbols as +, — , V, 0, 1, 2; each of these has its own proper
meaning.
/ adopt this fundamental idea of distinguishing two kinds of
symbols (which unfortunately is not strictly carried out in the
theory of magnitude - footnote of Frege's: Consider the
symbols 1, log, sin, Lim. -) in order to make it generally
applicable in the wider domain of pure thought. Accordingly,
I divide all the symbols I use into those that can be taken to
mean various things and those that have a fully determinate
sense. The first kind are letters, and their main task is to be
the expression of generality. For all their indeterminateness,
it must be laid down that a letter retains in a given context
the meaning once given to it.
2. Russell
42.12 The idea of a variable, as it occurs in the present
work, is more general than that which is explicitly used in
ordinary mathematics. In ordinary mathematics, a variable
generally stands for an undetermined number or quantity. In
mathematical logic, any symbol whose meaning is not deter-
minate is called a variable, and the various determinations of
which its meaning is susceptible are called the values of the
variable. The values may be any set of entities, propositions,
functions, classes or relations, according to circumstances.
If a statement is made about 'Mr A and Mr B\ 'Mr A' and
'Mr BJ are variables whose values are confined to men. A
variable may either have a conventionally-assigned range of
values, or may (in the absence of any indication of the range
of values) have as the range of its values all determinations
which render the statement in which it occurs significant. Thus
when a text-book of logic asserts that lA is A', without any
indication as to what A may be, what is meant is that any
statement of the form 'A is A' is true. We may call a variable
restricted when its values are confined to some only of those
of which it is capable ; otherwise we shall call it unrestricted.
Thus when an unrestricted variable occurs, it represents any
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PROPOSITIONAL LOGIC
object such that the statement concerned can be made
significantly (i.e. either truly or falsely) concerning that
object. For the purposes of logic, the unrestricted variable is
more convenient than the restricted variable, and we shall
always employ it. We shall find that the unrestricted variable
is still subject to limitations imposed by the manner of its
occurrence, i.e. things which can be said significantly con-
cerning a proposition cannot be said significantly concerning
a class or a relation, and so on. But the limitations to which
the unrestricted variable is subject do not need to be explic-
itly indicated, since they are the limits of significance of the
statement in which the variable occurs, and are therefore
intrinsically determined by this statement. This will be more
fully explained later.
To sum up, the three salient facts connected with the use
of the variables are: (1) that a variable is ambiguous in its
denotation and accordingly undefined; (2) that a variable
preserves a recognizable identity in various occurrences
throughout the same context, so that many variables can
occur together in the same context each with its separate
identity; and (3) that either the range of possible determina-
tions of two variables may be the same, so that a possible
determination of one variable is also a possible determination
of the other, or the ranges of two variables may be different,
so that, if a possible determination of one variable is given
to the other, the resulting complete phrase is meaningless
instead of becoming a complete unambiguous proposition
(true or false) as would be the case if all variables in it had
been given any suitable determinations.
F. TRUTH-VALUES
Truth-values, which are of great importance in formal logic,
form a special kind of value. The idea is already present in the
Megarian school (20.07), but its expression and first description
comes from Frege. His doctrine is linked to his own semantics,
according to which every proposition is a name for truth or falsity,
and in this he has not been generally followed, but the concept of
truth-value has been accepted by all.
We give first a text of Frege's :
42.13 But that indicates at the same time that the domain
of values for functions cannot remain limited to numbers ; for if
327
MATHEMATICAL VARIETY OF LOGIC
I take as arguments of the function £2 = 4 the numbers 0,1,2, 3,
in succession, I do not get numbers. '02 = 4', '12 = 4', '22 = 4',
'32 = 4'5 are expressions now of true, now of false thoughts.
I express this by saying that the value of the function £2 = 4
is the truth-value either of what is true or of what is false.
From this it can be seen that I do not intend to assert any-
thing by merely writing down an equation, but that I only
designate a truth-value; just as I do not intend to assert
anything by simply writing down '22' but only designate
a number. I say: The names '22 =4' and '3 >2' stand for the
same truth-value' which I call for short the True. In the same
manner '32 = 4' and '1 > 2' stand for the same truth-value,
which I call for short the False, just as the name '22' stands for
the number 4. Accordingly I say that the number 4 is the
reference of '4' and of '22', and that the True is the reference
of '3 > 2'. But I distinguish the sense of a name from its
reference. The names '22' and ' 2+ 2' have not the same sense,
nor have '22 = 4' and '2 + 2 = 4'. The sense of the name for a
truth-value I call a thought. I say further that a name expresses
is its sense, and what it stands for is its reference. I designate
by a name that which it stands for.
The function £2 = 4 can thus have only two values, the
True for the arguments + 2 and - 2 and the False for all other
arguments.
Also the domain of what is admitted as argument must be
extended - indeed, to objects quite generally. Objects stand
opposed to functions. I therefore count as an object everything
that is not a function: thus, examples of objects are numbers,
truth-values, and the ranges to be introduced further on. The
names of objects - or proper names - are not therefore accom-
panied by argument-places, but are 'saturated', like the
objects themselves.
42.14 I use the words, 'the function 0(5) has the same
range as the function Y(5)', to stand for the same thing as
the words, 'the functions 0(5) and T(£) have the same value
for the same arguments'. This is the case with the functions
52 = 4 and 3.£2 = 12, at least if numbers are taken as argu-
ments. But we can further imagine the signs of evolution and
multiplication defined in such a manner that the function
(52 = 4) = (3.£2 = 12) has the True as its value for any
argument whatever. Here an expression of logic may also be
used: 'The concept square root of 4 has the same extension as
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PROPOSITIONAL LOGIC
the concept something whose square when trebled makes 12\ With
those functions whose value is always a truth-value we can
therefore say 'extension of the concept' instead of 'range of the
function', and it seems suitable to say that a concept is a
function whose value is always a truth-value.
Independently of Frege, Peirce developed similar thoughts in
1885. His treatment of truth-values is more formalistic and not
tied to any particular semantic theory. However, this formalism
enabled him to formulate one which seems to qualify him to be
regarded as a precursor of many-valued logics.
42.15 According to ordinary logic, a proposition is either
true or false, and no further distinction is recognized. This
is the descriptive conception, as the geometers say; the metric
conception would be that every proposition is more or less
false, and that the question is one of amount. At present we
adopt the former view.
42.16 Let propositions be represented by quantities. Let
v and f be two constant values, and let the value of the quan-
tity representing a proposition be v if the proposition is true
and be f if the proposition is false. Thus, x being a proposition,
the fact that x is either true or false is written
(x - f) (v - x) = 0.
So
(x - f) (v - g) = 0
will mean that either x is false or y is true. . . .
42.17 We are, thus, already in possession of a logical
notation, capable of working syllogism. Thus, take the
premisses, 'if x is true, y is true', and 'if y is true, z is true'.
These are written
(x - f) (v - g) = 0
(y - f) (v - z) = 0.
Multiply the first by (v - z) and the second by [x - f) and add.
We get
(x - I) (v - f) (v - z) = 0,
or dividing by v - f, which cannot be 0,
(x - f) (v - z) = 0;
and this states the syllogistic conclusion, 'if x is true, z is
true'.
42.18 But this notation shows a blemish in that it express-
es propositions in two distinct ways, in the form of quan-
tities, and in the form of equations; and the quantities are
329
MATHEMATICAL VARIETY OF LOGIC
of two kinds, namely those which must be either equal to
f or to v, and those which are equated to zero. To remedy
this, let us discard the use of equations, and perform no
operations which can give rise to any values other than f
and v.
42.19 Of operations upon a simple variable, we shall need
but one. For there are but two things that can be said about
a single proposition, by itself; that it is true and that it is
false,
x = v and x = f.
The first equation is expressed by x itself, the second by any
function, 9, of x, fulfilling the conditions
cpv = f <pf = v.
This simplest solution of these equations is
yx = i + v - x.
G. TRUTH-MATRICES
1. Peirce
The standpoint revealed in the last text comes near to defining
propositional functors by means of truth-values. Tabular definitions
of this kind have already been met with in the Stoic-Megarian school
(20.07 ff.), later on in Boole, though without explicit reference to
truth-values, and finally in Frege's Begriffsschrift (41.12). Peirce has
the notion quite explicitly, and in connection, moreover, with
ancient logic, in 1880:
42.20 There is a small theorem about multitude that it
will be convenient to have stated, and the reader will do well
to fix it in his memory correctly. ... If each of a set of m
objects be connected with some one of a set of n objects, the
possible modes of connection of the sets will number nm.
Now an assertion concerning the value of a quantity either
admits as possible or else excludes each of the values v and f.
Thus, v and f form the set m objects each connected with one
only of n objects, admission and exclusion. Hence there are,
nm , or 22, or 4, different possible assertions concerning the
value of any quantity, x. Namely, one assertion will simply
be a form of assertion without meaning, since it admits either
value. It is represented by the letter, x. Another assertion
will violate the hypothesis of dichotomies by excluding both
values. It may be represented by x. Of the remaining two,
one will admit v and exclude f, namely x; the other will
admit f and exclude v, namely x.
330
PROPOSITIONAL LOGIC
Now, let us consider assertions conce-
rning the values of two quantities, xand
y. Here there are two quantities, each
of which has one only of two values; so
that there are 22, or 4, possible states of
things, as shown in this diagram.
Above the line, slanting upwards to
the right, are placed the cases in which
x is v; below it, those in which x is f.
Above the line but slanting downward
to the right, are placed the cases in which y is v; below it,
those in which y is f. Now in each possible assertion each of
these states of things is either admitted or excluded ; but not
both. Thus m will be 22, while n will be 2; and there will
be nm, or 22, or 16, possible assertions. . . .
Of three quantities, there are 23, or 8, possible sets of
values, and consequently 28, or 256, different forms of propo-
sitions. Of these, there are only 38 which can fairly be said to
be expressible by the signs [used in a logic of two quantities].
It is true that a majority of the others might be expressed
by two or more propositions. But we have not, as yet, expressly
adopted any sign for the operation of compounding propo-
sitions. Besides, a good many propositions concerning three
quantities cannot be expressed even so. Such, for example,
is the statement which admits the following sets of values :
x y z
V
V V
V
f f
i
V f
i
f V
Moreover, if we were to introduce signs for expressing
[each of] these, of which we should need 8, even allowing the
composition of assertions, still 16 more would be needed to
express all propositions concerning 4 quantities, 32 for 5,
and so on, ad infinitum.
2. Wittgenstein
The same doctrine was systematically elaborated about 1920 by
J. Lukasiewicz, E. L. Post and L. Wittgenstein. We give the relevant
text from the last:
42.21. With regard to the existence of n atomic facts there
331
MATHEMATICAL VARIETY OF LOGIC
are
Kn = 2 ( ) possibilities.
It is possible for all combinations of atomic facts to exist,
and the others not to exist.
42.22 To these combinations correspond the same number
of possibilities of the truth - and falsehood - of n elementary
propositions.
42.23 The truth-possibilities of the elementary propositions
mean the possibilities of the existence and non-existence of
the atomic facts.
42.24 The truth-possibilities can be presented by schemata
of the following kind ('T' means 'true, lF' 'false'. The rows
of T's and F's under the row of the elementary propositions
mean their truth-possibilities in an easily intelligible sym-
bolism).
P q r
T T T
F T T
~T~ F _T
T T F
F~ F ~f
F T F
T F F
F F F
p <L
T T
F T
T F
F F
42.25 . . . The truth-possibilities of the elementary propo-
sitions are the conditions of the truth and falsehood of the
propositions.
42.26 It seems probable even at first sight that the intro-
duction of the elementary propositions is fundamental for
the comprehension of the other kinds of propositions. Indeed
the comprehension of the general propositions depends palpably
on that of the elementary propositions.
42.27 With regard to the agreement and disagreement of a
proposition with the truth-possibilites of n elementary
Kn (K \
propositions there are 2 ( n) = Ln possibilities.
42.28 . . . Thus e.g.
332
PROPOSITIONAL LOGIC
p
q
T
T
T
F
T
T
r
F
F
F
T
is a prepositional sign.
42.29 . . . Among the possible groups of truth-conditions
there are two extreme cases.
In the one case the proposition is true for all the truth-
possibilities of the elementary propositions. We say that the
truth-conditions are tautological.
In the second case the proposition is false for all the truth-
possibilities. The truth-conditions are self-contradictory.
In the first case we call the proposition a tautology, in the
second case a contradiction.
42.30 The proposition shows what it says, the tautology
and the contradiction that they say nothing.
The tautology has no truth-conditions, for it is uncondi-
tionally true; and the contradiction is on no condition true.
Tautology and contradiction are without sense.
(Like the point from which two arrows go out in opposite
directions.)
(I know, e.g. nothing about the weather, when I know
that it rains or does not rain.)
42.31 Tautology and contradiction are, however, not
nonsensical; they are part of the symbolism, in the same
way that '0' is part of the symbolism of Arithmetic.
The name 'tautology' and the last quotation show the peculiar
(extremely nominalist) tendency underlying Wittgenstein's semantic
views. It is diametrically opposed to Frege's tendency and from his
point of view misleading.
H. DECISION PROCEDURE OF LUKASIEWICZ
The tables of values constructed in the texts just cited provide a
decision procedure for propositional functions, i.e. a procedure which
enables one to decide whether a function is a logical law (whether it
becomes a true proposition for every correct substitution). The basic
idea of such a procedure is present in Schroder (42.32). It was
developed by E.L. Post (42.33) and was known to J. Lukasiewicz at
the same time (42.34). It is set out in full in the manuals of Hilbert
333
MATHEMATICAL VARIETY OF LOGIC
and Ackermann (42.35), 1928, and T. Kotarbinski, 1929. We quote
Kotarbinski's text, because of its clarity. The author writes 'p" for
'not p', and uses '+', '<', ' = ', as signs of addition, implication and
equivalence respectively.
42.36 We shall now give a very simple method of verifica-
tion for the propositional calculus, which enables one to
verify the correctness of every formula in this domain [viz.
the zero-one method of verification]. We stipulate for this
purpose that it is permitted to write, say, zero for a false
proposition, and one for a true. With the help of this sym-
bolism we now investigate whether a given formula becomes
a true proposition for all substitutions of propositions for
propositional variables - always under the condition that the
same (proposition) is substituted for the same (variable), or
whether on the other hand it becomes a false proposition for
some substitutions. In the first case it is a valid formula, in
the second an invalid one. . . .
We recollect in this connection: (1) that the negation of a
true proposition is always a false proposition, and conver-
sely; (2) that the logical product is true only when both
factors are true; (3) that the logical sum is always true when
at least one of its parts is true, and false only in the case
that both its parts are false; (4) that implication is false
only when its antecedent is true and its consequent false;
(5) that equivalence is true only when both sides are true or
both false; but when one side is true, the other false, then the
whole equivalence is false. If, for example, we put zero for p
and one for q in a formula, then this formula can be further
simplified by writing a zero in place of the product of p and q
wherever this occurs throughout the formula- and analogously
in the case of the other functions. . . .
If, after the application of all possible substitutions of
zero and one for propositional variables in a given formula,
and after carrying out the . . . simplifications described, the
formula always reduces to a one, then it is true. But if it
is zero for even a single choice of substitutions, it is invalid.
To have an example, we verify the formula of transpo-
sition . . .
(p < q) = W < p')
1. We suppose that true propositions have been substituted
both for p and for q. Our formula then takes on the form
(i < i) = (r < i').
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PROPOSITIONAL LOGIC
Simplifying it by Rule (1), we get:
(1<1) = (0<0)
. . . further ... we get
1 = 1
which ... we can replace by
1.
The process is then repeated with the other three possible
substitutions.
§43. PROPOSITIONAL LOGIC AS A SYSTEM
In the last two paragraphs we have spoken of the basic concepts
and one of the main methods of modern propositional logic. Now we
come on to show the second, axiomatic, method at work, in some
sample sections reproduced from different propositional systems.
These sections will be taken from the systems of McColl (1877),
Frege (1879 and 1893), Whitehead and Russell (1910) - here we insert
a text from Peirce, connected with the Sheffer stroke - and finally
Lukasiewicz (1920). In this last system, two-valued propositional
logic seems to have reached the term of its development.
A. McCOLL
The ensuing texts, which are a continuation of the definitions in
41.05 ff., contain rules for an algebraic system of propositional logic,
constructed in the spirit of Boole. It may be compared with the
non-algebraic system of Lukasiewicz (43.45). It may be remarked
that while this algebraic style has been for the most part superseded,
there have also been quite recent algebraic systems (Tarski).
43.01 Rule 1. - The rule of ordinary algebraical multi-
plication applies to the multiplication of indeterminate
statements, thus :
A{B + C) =AB + AC; (A + B) (C + D) = AC + AD + BC +
BD;
and so on for any number of factors, and whatever be the
number of terms in the respective factors.
43.02 Rule 2. - Let A be any statement whatever, and let
B be any statement which is implied in A [and which must
therefore be true when A is true, and false when A is false];
or else let B be any statement which is admitted to be true
independently of A; then [in either case] we have the equation
A = AB. As particular cases of this we have A = AA =
335
MATHEMATICAL VARIETY OF LOGIC
AAA = etc., as repetition neither strengthens nor weakens
the logical value of a statement. Also,
A =A{B+B')=A{B+B'){C+C')=etc., for
B+B' = l=C+C'=etc. [see Def. 4] (41.08).
43.03 Rule 3. - {ABy=AB,+AfB+A,B'
=AB'+A,(B+B')=AB,+A'
=A' B+B'(A+A')=A' B+B' ,
for A+A' — l and B + B' = 1. Similarly we may obtain various
equivalents (with mutually inconsistent terms) for (ABC)',
(ABCD)', etc.
43.04 Rule 4. - [A+R)' = A' B' ; {A+B+C)' = A'B'C ; and
so on.
43.05 Rule 5. -A+B ={(A+B)'}' = {A'B')'
=AB,+(A,+A)B=AB'+B
=A'B+A(B'+B = A'B+A.
Similarly we get equivalents (with mutually inconsistent
terms) for A+B+C,A+B+C+D, etc.
43.06 Rule 11. - If A:B, then B':A'. Thus the implica-
tions A:B and B':A' are equivalent, each following as a
necessary consequence of the other. This is the logical prin-
ciple of 'contraposition'.
43.07 Rule 12. - If A:B, then AC:BC, whatever the state-
ment C may be.
43.08 Rule 13. - If A:*, B:$, C:y, then ABC: a^y, and so
on for any number of implications.
43.09 Rule 14. - If AB=0, then A:B' and B:A' .
43.10 Definition 13. - The symbol A+B asserts that A does
not imply B; it is thus equivalent to the less convenient
symbol (A: By.
43.11 Rule 15. - If A implies B and B implies C, then A
implies C.
43A2 Bute 16. - If A does not imply B7 then B' does not
imply A'; in other words, the non-implications A+B and
B'+Af are equivalent.
43.13 Rule 17. - If A implies R but does not imply C, then
B does not imply C in other words, from the two premises
A:B and A+C, we get the conclusion B+C.
43.14 The following formulae are all either self-evident
or easily verified, and some of them will be found useful in
abbreviating the operations of the calculus : -
(1) i< = 0,0' = l;
(2) l = l+a = l+a + b = l+a + b + c, etc. ;
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PROPOSITIONAL LOGIC
(3) {ab + a'b')' = a'b + ab'
(a'b + ab')' = ab + a'b'
(4
(5
(6
(V
(8
(10
(11
(12
(13
(14
a : a + b : a + b + c, etc. ;
(a + A) (a+ D) (a + C) ... = a + A£C ...;
(a : b) : a' + b;
(a = b) = (a ib) {b : a);
(a — b) : ab + a'b' ;
(A :a) (5:6) (C : c) ... : (ABC ... : afo ...);
(A : a) ( £ : b) ( C: c) ... : [A + £ + C ...:a + b + c +
...);
(A :aj) [B:x) {C : x) ... = (A + B + C + ... :x);
(x:A) {x: B) (as C) ... = (x:i5C ...);
(A :aj) + (B:«) + (C : x) + ... : {ABC ... :a?);
(x : A) + [x : B) + (x : C) + ... : (x : A + B + C + ...).
B. FREGE'S RULES OF INFERENCE
One of Frege's most important intuitions concerned the distinc-
tion between theorems and rules of inference. This is already to be
found in the Begriffsschrift (43.15), where he uses only a single
rule; in the Grundgesetze he adopts several for reasons of practical
convenience, of which we give four:
43.16 // the lower member of a proposition differs from a
second proposition only in lacking the judgment-stroke, one
can conclude to another proposition which results from the first
by suppression of that lower member.
'Lower member', i.e. antecedent. The sense is therefore: Given
' | — if B, then A' and further ' | — B\ then we may suppress B in
the conditional proposition to obtain ' | — A\ This is the modus
ponendo ponens (22.04).
43.17 A lower member may be exchanged with its upper
member, if at the same time the truth-value of each is changed.
Thus, given 'if B, then A\ one may write 'if not A, then not B' ;
this is the rule of simple contraposition (31.17, cf. 43.22 [28]).
43.18 // the same combination of symbols occurs as upper
member in one proposition and lower member in another, one
can conclude to a proposition in which the upper member of
the second appears as upper member, and all lower members of
the two, save the one mentioned, as lower members. Bui lower
members which occur in both, need only be written once.
337
MATHEMATICAL VARIETY OF LOGIC
Given 'if C, then B' and 'if B, then A\ one may write 'if C, then
A\ This is the rule corresponding to the law of syllogism (cf. 31.18).
43.19 // two propositions correspond in their upper members,
while a lower member of one differs from a lower member of the
other only in respect of a preceding negation-stroke, then we
can conclude to a proposition in which the corresponding upper-
member appears as upper member, and all lower members of the
two, with the exception of the two mentioned, as lower members.
Lower members which occur in both are only to be written down
once.
Frege's concrete example (43.20) is this: Given 'if e, then if not d,
then: if b, then a' and 'if e, then if d, then: if b, then a', one may
write: 'if e, then: if b, then a'.
Lukasiewicz, deriving from Frege, formulates the difference
between thesis and rule, and states the most important rule, as
follows :
43.21 A logical thesis is a proposition in which besides
logical constants there occur only propositional or name-
variables and which is true for all values of the variables that
occur in it. A rule of inference is a direction which empowers
the maker of inference to derive new theses on the basis of
already admitted theses. Thus e.g. the laws of identity given
above are logical theses, while the following 'rule of detach-
ment' is a rule of inference :
Whose admits as true the implication 'if a, then p' and the
antecedent 'a' of this implication, has the right to admit as
true also the consequent '[}' of this implication.
Thus for Lukasiewicz, 'logical thesis' covers both axioms and
derived propositions.
C. PROPOSITIONAL LAWS FROM THE B EGR I FFSSCH R IFT
Space prevents us from giving Frege's propositional schemata
(corresponding to Lukasiewicz's theses) in the original symbolism;
Instead, we translate some of them into Peano-Russellian.
43.22 01. ud .bD a
02. cD.bDaiDicDb.D.cDa
03. bDa.Di.cD.bDaiDicDb.D.cDa
04. b d a . d : c d . b d a : . d : . b d a : d : c d b . d . c
d a
05. bDaiDicDb.D.cDa
338
(x:
'"(i-;::r"Mii-:)
T'"ii:::r"'(v:i)
a re
a ~r
a ^w
a - c~/>
An example of Frege's Symbolism taken from "Begriffsschrift", p. 56.
PROPOSITIONAL LOGIC
06. c d . b d a : . d : . c : d : d d b . d . d d a
07. b d a : . d : . d d . c d b : d : d . d . c d a
08. d d . b d a : d : b . d . d d a
09. c d b : d : b d a . d . c d a
10. e d . d d b : d a : . d . d d . e d b : d a
11. CDb.D.aiD.bDCl
12. d 3 : c . d . 6 d a : . d : . d : d : b .
13. d d : c . d . b d a : . d : . b : d : d .
14. n:.rf:D:c.D.ba::D::e
d . d . c d a
15. m.diDic.DiDo:::::!*:.::
d . c d a
d . c d a
d . c d a
: . D : . 6 : d
::>:</
16.
e d : . d d : c d .b d a: : d : : e d : . d d : b d
17.
dD:c.D.bDa:.D:.cD:b.D.dDa
18.
cD.bDaiDi.dDciDibD.dDa
19.
dD.CDblDl.bDdlDldD.CDa
20.
e d : d d . c d b : : d : : b d a: . d : .e d : d . :
21.
dDb.DdlDl.dDC.DlCDb.DO.
22.
f d : : eD : . d d : c d .b d a:: . d :: . f :> : : <
d : b d . c d a
23.
d d : c d . b d a : . d : : e d d . d : . c d : b . z
24.
CDd.DlCD.bDCl
25.
dD.CDdlDl.dDlCD.bDa
26.
b d . a d a
27.
a d a
28.
bDa.D.~aD~b
29.
cD.bDa:D:cD.~aD~b
30.
bD.cDa\D.CD.~aD~b
31.
~ ~ a d a
32.
~bDa,D.~aD~~b:Di~bDa.:
Db
33.
~bDa.D.~aDb
34.
cD.~bDa:D:cD.~aDb
35.
cD.~bDa:D:~aD.cDb
36.
a d . <— ' a d b
37.
~CDb.Da:D.cDa
38.
~ a . d . a d b
39.
~aDa.D.~aDb
40.
~bD:~aDa.Da
41.
a d ~ ~ a
42.
~ ~ (a d a)
43.
~ a d a . d . a
c d a
c d a
e d : . d
e d a
339
MATHEMATICAL VARIETY OF LOGIC
44. ~aDc.D:cDa.Da
45. ~cDa.D.~aDc\D'. .~cDa:D:cDa.Da
46. ~cDa.D:cDa.D.a
47. ~cDb.D:.bDa.D:cDa.Da
48. d d . ~ c d b : d : . b d a . d : c d a . d . d d a
49. ~cDb.D\.cDa.D:bDa.Da
50. CDa.D:.ha.D:~CDfi.Dfl
51. d d . c d a: d ::b d a . d : . d d : ~ c d b . d a
52. C ^d.D.f{c)Df{d)
53. / (c) d : c = d . d f (d)
54. c = c
55. c = d . d . d = c
56. d=c.D.f(d)Df{c)
57. c =d.D .f{d)Df{c)
D. WHITEHEAD AND RUSSELL
Passing over Peano, we come now to the Principia Malhematica
of Whitehead and Russell (Vol. 1, 1910).
1. Primitive symbols and definition
Besides variables, Frege's sign of assertion V and Peano's dots
and brackets, the Principia uses only two undefined primitive
symbols: '~' and 'v'. 'p' is read as 'not p', 'p v q' as 'p or q' the
alternation being non-exclusive (43.23).
Implication is defined :
43.24 *1.01. p d q . = . ~ p v q Di.
2. Axioms (Primitive Propositions)
43.25 *1.1. Anything implied by a true elementary propo-
sition is true. Pp. (Footnote : The letters "Pp" stand for "primi-
tive proposition", as with Peano.)
The above principle ... is not the same as "if p is true, then if
p implies q, q is true". This is a true proposition, but it holds
equally when p is not true and when p does not imply q.
It does not, like the principle we are concerned with, enable
us to assert q simply, without any hypothesis. We cannot
express the principle symbolically, partly because any
symbolism in which p is variable only gives the hypothesis that
p is true, not the fact that it is true.
43.26 *1.2. h :pv p. Dp Pp.
This proposition states : "If either p is true or p is true, then
340
PROPOSITIONAL LOGIC
p is true". It is called the "principle of tautology", and will be
quoted by the abbreviated title of "Taut". It is convenient
for purposes of reference, to give names to a few of the more
important propositions; in general, propositions will be refer-
red to by their numbers.
43.27 *1.3. h :q. d . p v q Pp.
This principle states: "If q is true, then 'p or 7' is true".
Thus e.g. if q is "to-day is Wednesday" and p is "to-day is
Tuesday", the principle states: "If to-day is Wednesday, then
to-day is either Tuesday or Wednesday". It is called the
"principle of addition". . . .
43.28 *1.4. [- : p v q . d . q v p Pp.
This principle states that "p or q" implies llq or p". It states
the permutative law for logical addition of propositions, and
will be called the "principle of permutation". . . .
43.29 *1.5. npv(gvr).3.?v(pvr)Pp.
This principle states: "If either p is true, or 'q or r' is true,
then either q is true, or 'p or r' is true". It is a form of the
associative law for logical addition, and will be called the
'associative principle'. . . .
43.30 *1.6. \- :: . q d r . d : p v q . d . p v r Pp.
This principle states: "If q implies r, then 'p or q' implies
*p or p' ". In other words, in an implication, an alternative
may be added to both premiss and conclusion without
impairing the truth of the implication. The principle will be
called the "principle of summation", and will be referred to
as "Sum".
3. Statement of proofs
Two examples will explain the method of proof used in the
Principia:
43.31 *2.02. 1- :q.D . p Dq
Dem.
< — ' p
Add \- : q . d . ~ pv q (1)
P
(1) .(*1.01) h :q. D.p Dq
This is to be read: take 'Add', i.e. 43.27:
q. D.p v q
and in it substitute '~ p' for 'p'; we obtain
341
MATHEMATICAL VARIETY OF LOGIC
As according to 43.24 '~ p v q' and 'p d q' have the same meaning,
the latter can replace the former in ( 1 ) , which gives the proposition to
be proved; it corresponds to the Scholastic verum sequitur ad quod-
libel
That proof is a very simple one; a slightly more complicated
example is:
43.32 *2.3. h :p v (q v r) . d . p v (r v q)
Dem.
Perm
Sum
q v r, r v q
h : q v r • d . r v q:
d t- : p v (q v r) . d .p v (r v q)
4. Laws
in
the
43.33 The most important propositions proved
present number are the following : . . .
*2.03. \-:pD~q.D.qD~p
*2.15. \-:~pDq.D.~qDp
*2.16. \-:pDq.D.~qD~p
*2.17. h:~p^p.:.p:^
These four analogous propositions consitute the "prin-
ciple of transposition" . . .
*2.04. h :. p . d . q d r : d : q . d . p d r
*2.05. h i.qDr.DipDq.D.pDr
*2.06. h i.pDq.DiqDr.D.pDr
These two propositions are the source of the syllogism in
Barbara (as will be shown later) and are therefore called "the
principle of the syllogism" . . .
*2.08. \- . pD p
I.e. any proposition implies itself. This is called the "prin-
ciple of identity" . . .
*2.21. \- : ~ p . d . p d q
I.e. a false proposition implies any proposition.
Next the Principia gives a series of laws concerning the logical
product (43.34). At their head stand the two definitions:
43.35 *3.01. p . q . = . — [~ p v — q) Df
where "p . q" is the logical product of p and q.
*3.02. pDqDr. = .pDq.qDrDf.
This definition serves merely to abbreviate proofs.
43.36 The principal propositions of the present number are
the following:
342
PROPOSITIONAL LOGIC
*3.2. h :. p , d : q . d . p . q
I. e. "p implies that q implies p . ry", i.e. if each of two pro-
positions is true, so is their logical product.
*3.26. \- : p .q . d . p
*3.27. h : p . q . p . q
I.e. if the logical product of two propositions is true, then
each of the two propositions severally is true.
*3.3. h :. p . q . d . r : d : p . d . q d r
I.e. if p and q jointly imply r, then p implies that q implies
r. This principle (following Peano) will be called "exportation",
because q is "exported" from the hypothesis . . .
*3.31. h :. p . d . q d r : d : p . q . d . r . . .
*3.35. I- : p . 
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